Method and system for digital signal transmission

ABSTRACT

The invention relates to a method and an arrangement for transmitting a digital signal consisting of symbols, which arrangement comprises a coder ( 308 ) for coding complex symbols to channel symbols in blocks having the length of a given K, means ( 312 ) for transmitting the channel symbols via several different channels and two or more antennas ( 314  to  318 ). The coder ( 308 ) is arranged to code the symbols using a code matrix, which can be expressed as a sum of 2K elements, in which each element is a product of a symbol or symbol complex conjugate to be transmitted and a N×N representation matrix of a complexified anti-commutator algebra, extended by a unit element, and in which each matrix is used at most once in the formation of the code matrix. A code matrix is also provided which is formed by matrices of a portion of the symbols placed on the diagonal of the code matrix and by matrices of a second portion of symbols along the anti-diagonal of the code matrix.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. Section 119 based onFinnish Application Serial Number 20000406 filed on Feb. 22, 2000, whichhas been filed as co-pending U.S. application Ser. No. 10/225,457 andFinnish Application Serial Number 20001944 filed on Sep. 4, 2000, whichhas been filed as co-pending U.S. application Ser. No. 10/378,068. Thisapplication is also related to and claims priority to U.S. ProvisionalApplication No. 60/193,402 filed on Mar. 29, 2000, entitled “CLASS OFSPACE-TIME BLOCK CODES FOR MORE THAN TWO ANTENNAS” all assigned toassignee of present application and all incorporated herein byreference.

FIELD OF THE INVENTION

This invention relates generally to methods and systems for achievingtransmit diversity in a telecommunication system. More particularly, thepresent invention relates to an apparatus and associated method forusing a space-time block code to reduce bit error rates of a wirelesscommunication in a spread spectrum receiver.

Mathematical Foundations

The review of mathematical tools, nomenclature, and notations followswhich is intended to be used for an understanding of the presentinvention. Multiple notations may be used in same parts of theapplication to better describe the invention. Other mathematicaltechniques, nomenclature, and notations may be used to describe thepresent invention without departing from the spirit and scope thereof.

A vector may be represented in Dirac notation proposed by P.A.M. Dirac:

-   -   Bra a| which may be represented by a row matrix or        α_(i)=[α₁,α₂,α₃]    -   Ket |b which may be represented by a column matrix        ${\beta_{i} = \begin{bmatrix}        \beta_{1} \\        \beta_{2} \\        \beta_{3}        \end{bmatrix}};$        and    -   C is defined as        $\left. {= {{\sum\limits_{i}l}❘i}} \right\rangle\left\langle i \right.$

Thus BraCKet notation is an analogy with the dot product${A \cdot B} = {\sum\limits_{i}{A_{i}{B_{i}.}}}$

Definition: The inner product of two vectors in a vector space V is acomplex number, such thata|b=b|a*a|(β|b+γ|c)=βa|b+γa|ca|a≧0, and a|a=0iff|a=0.

Therefore, the inner product may be defined in summation notation as$\left\langle {{{a\left. b \right\rangle} \equiv {{\alpha_{1}^{*}\beta_{1}} + {\alpha_{2}^{*}\beta_{2}} + \ldots + {\alpha_{n}^{*}\beta_{n}}}} = {\sum\limits_{i = 1}^{n}{\alpha_{i}^{*}{\beta_{i}.}}}} \right.$

A state vector may be represented as a linear combination with suitablecoefficients of a set of base vectors I, j:$\left. {{\left. {\left\langle {{i\left. a \right\rangle} = \left. C_{1}\Leftrightarrow \right.} \right.a} \right\rangle = {\sum\limits_{i}{\left. i \right\rangle C_{i}}}}{\left\langle {{i\left. b \right\rangle} = \left. D_{1}\Leftrightarrow \right.} \right.b}} \right\rangle = {\sum\limits_{i}{\left. i \right\rangle D_{i}}}$$\left\langle {{a\left. i \right\rangle} = \left\langle {{i\left. a \right\rangle^{*}\left\langle b \right.} = {\sum\limits_{j}{D_{j}^{*}\left\langle j \right.\left\langle {{b\left. a \right\rangle} = {\sum{D_{j}^{*}\left\langle {j{i}} \right\rangle C_{i}}}} \right.}}} \right.} \right.$

-   -   but        $\left. {\left\langle i \right.j} \right\rangle = {\delta_{ij} \equiv \begin{Bmatrix}        1 & {{{if}\quad i} = j} \\        0 & {{{if}\quad i} \neq j}        \end{Bmatrix}}$        (the Kronecker delta δ_(ij)) is a symbol that is defined to be 0        for i≠j and to be 1 when i=j and can be represented by the unit        matrix $\begin{pmatrix}        1 & 0 \\        0 & 1        \end{pmatrix}\quad$        which may also be referred to as the identity matrix I₂ or for        an N dimension basis I_(N). The vectors i and j are normalized        and form an orthonormal basis. Sometimes {|e_(i)}_(i=1) ^(N) is        used as a basis in a N-dimensional vector space.

Vectors |a,|bεV are orthogonal if a|b=0.

Operators can be represented as${A_{ij} \equiv \left\langle {i{A}j} \right\rangle} = \begin{bmatrix}A_{11} & A_{12} & \ldots & \ldots & A_{1j} \\A_{21} & A_{22} & \ldots & \ldots & A_{2j} \\\vdots & \vdots & ⋰ & \vdots & \vdots \\\vdots & \vdots & \vdots & ⋰ & \vdots \\A_{i1} & A_{i2} & \ldots & \ldots & A_{ij}\end{bmatrix}$

-   -   |b=A|a operator A performs operation on vector state |a to        produce vector state |b,        $\left. {\left\langle i \right.b} \right\rangle = {\sum\limits_{j}{\left\langle {i{A}j} \right\rangle\left\langle {j{\left. a \right\rangle.}} \right.}}$    -   b|A|a*=a|A^(H)|b where A^(H) is an operator whose matrix        elements are A_(ij) ^(H)=(A_(ji)) and is called Hermitian when        A^(H)=A.

A system may be represented in terms of the unit matrix and a set ofmatrices: $M = {\begin{bmatrix}a & b \\c & d\end{bmatrix} = {{a\begin{pmatrix}1 & 0 \\0 & 0\end{pmatrix}} + {b\begin{pmatrix}0 & 1 \\0 & 0\end{pmatrix}} + {c\begin{pmatrix}0 & 0 \\1 & 0\end{pmatrix}} + {d\begin{pmatrix}0 & 0 \\0 & 1\end{pmatrix}}}}$

An example of such a set of matrices is called Pauli-Spin matrices:${\sigma_{z} = \begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}},{\sigma_{x} = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{{{and}\quad\sigma_{y}} = {\begin{pmatrix}0 & {- i} \\i & 0\end{pmatrix}.}}$  σ_(x) ²=1,σ_(y) ²=1,σ_(z) ²=1. σ_(x)σ_(y)=−σ_(y)σ_(x)=iσ _(z), σ_(y)σ_(z)=−σ_(z)σ_(y) =iσ _(x), and σ_(z)σ_(x)=−σ_(x)σ_(z)=iσ _(y).

The present application will also use the mathematical methods of grouptheory and algebras. Simply, a group is a set G of elements g togetherwith a binary operation called multiplication with the properties:∃ε∈G ε*g=g*ε=g and ∀g∈G,∃g ⁻¹ g*g ⁻¹ =g ⁻¹ *g=ε

-   -   which reads in other words: there exist an unique element ε of        the group G such that ε operating on g equals g operating on ε        equals g (the element ε is referred to as the identity); and for        every element g belonging to group G, there exists an element        g⁻¹ (called the inverse) such that g operating on g-inverse        equals g-inverse operating on g equals the identity ε.

A set of linearly independent vectors may be represented as groupelements, a vector space may be generated by taking a linear combinationof the elements of the set. The product already defined on the basis setcan be extended to all the elements of the vector space by linearitydefining a group algebra. The Pauli-Spin matrices may be thought of asgroup elements.

BACKGROUND OF THE INVENTION

As wireless communication systems evolve, wireless system design hasbecome increasingly demanding in relation to equipment and performancerequirements. Future wireless systems, which will be third generation(3G) systems and fourth generation systems compared to the firstgeneration analog and second generation digital systems currently inuse, will be required to provide high quality high transmission ratedata services in addition to high quality voice services. Concurrentwith the system service performance requirements will be equipmentdesign constraints, which will strongly impact the design of mobileterminals. The third and fourth generation wireless mobile terminalswill be required to be smaller, lighter, more power-efficient units thatare also capable of providing the sophisticated voice and data servicesrequired of these future wireless systems.

Time-varying multi-path fading is an effect in wireless systems wherebya transmitted signal propagates along multiple paths to a receivercausing fading of the received signal due to the constructive anddestructive summing of the signals at the receiver. This occursregardless of the physical form of the transmission path, (i.e. whetherthe transmission path is a radio link, an optical fiber or a cable).Several methods are known for overcoming the effects of multi-pathfading, such as time interleaving with error correction coding,implementing frequency diversity by utilizing spread spectrumtechniques, transmitter power control techniques, and the like. Each ofthese techniques, however, has drawbacks in regard to use for third andfourth generation wireless systems. Time interleaving may introduceunnecessary delay, spread spectrum techniques may require largebandwidth allocation to overcome a large coherence bandwidth, and powercontrol techniques may require higher transmitter power than isdesirable for sophisticated receiver-to-transmitter feedback techniquesthat increase mobile station complexity. All of these drawbacks havenegative impact on achieving the desired characteristics for third andfourth generation mobile terminals. Diversity is another way to overcomethe effects of multi-path fading.

Antenna diversity is one type of diversity used in wireless systems. Inantenna diversity, two or more physically separated antennas are used toreceive a signal, which is then processed through combining andswitching to generate a received signal. A drawback of diversityreception is that the physical separation required between antennas maymake diversity reception impractical for use on the forward link in thenew wireless systems where small mobile station size is desired.

A second technique for implementing antenna diversity is transmit (Tx)diversity. In transmit (Tx) diversity, a signal is transmitted from twoor more antennas and then processed at the receiver by using maximumlikelihood sequence estimator (MLSE) or minimum mean square error (MMSE)techniques. Transmit diversity has more practical application to theforward link in wireless systems in that it is easier to implementmultiple antennas in the base station than in the mobile terminal. FIG.1 is an illustration showing an example of a transmit diversity system.Transmit (Tx) diversity system 100 comprises multiple transmit antennas(Tx₁. . . Tx_(n); n=1 to N) 110, multiple channels (h₁ . . . h_(K)) 120where h_(k)=α_(k)e^(jθ) ^(k) ∀k=1 to K, and receiving antenna Rx 130.Channel interference may be created by structures or other features 140among various channel paths.

Use of Space-Time Block Code (STBC) aka Space-Time Code (STC) may beconsidered as diversity-creating. In space-time block code design, theessential design criteria are the achieved diversity, the rate of thecode, and the delay. Diversity is characterized by the number ofindependently decodable channels. For full diversity, this equals thenumber of transmit antennas. The rate of the code is the ratio of thespace-time coded transmission rate to the rate of an one-antennatransmission (i.e. the ratio of the transmission rate of the block codeto transmission rate of the uncoded scheme). The delay is the length ofthe space-time block code frame. Depending on the underlying modulationscheme, space-time block codes may be divided into real and complexcodes. Rate in the context of space-time block code may be understood asinefficiency in use of the antenna resources, leading to dilution ofmaximal bit-rates as compared to the inherent capacity of the underlyingwireless system specifications. This inefficient use of antennaresources may give rise to fluctuating of transmit powers. Suchfluctuation of transmit powers is referred to as the power-unbalanceproblem. Thus, the aims in the design of space-time block codes is toachieve unit rate (R=1) in order to use antenna resources as efficientlyas possible. The aim of diversity is to achieve maximal diversity.

Transmit diversity for the case of two antennas (N=2) is well studied.Both open-loop and closed-loop transmit diversity methods have beenunder consideration for 3G Wide-band Code Division Multiple Access(WCDMA) system.

An example of an open-loop concept is provided by Alamouti, who proposeda method of transmit diversity employing two antennas that offers secondorder diversity for complex valued signals. (S. M. Alamouti, “A SimpleTransmit Diversity Technique for Wireless Communications,” IEEE Journalon Selected Areas of Communications, vol. 16, no. 8, pp. 1451-1458,October 1998 and publication WO 99/14871 of ALAMOUTI et al. entitled“Transmitter Diversity Technique for Wireless Communication”) TheAlamouti method employs two transmit antennas, has a rate R=1 and amaximum-ratio diversity-combining detector.

The method achieves effective communication by encoding symbols thatcomprise negations and conjugation of symbols (i.e. negation ofimaginary parts) and simultaneously transmitting two signals (K=2) fromtwo antennas (N=2) to a receiving antenna (Rx) during a symbol period(T) also referred to as a time epoch. During one symbol period(T₁=t₀+T), the signal transmitted from a first antenna (Tx₁) is denotedby s₁ and the signal transmitted from the second antenna (Tx₂) isdenoted by s₂.where s₁ and s₂ are complex numbers. During the nextsymbol period (T₂=T₁+T), the signal−s₂*(negation/conjugate) istransmitted from the first antenna (Tx₁) and the signal s₁* (conjugate)is transmitted from the second antenna (Tx₂), where * is the complexconjugate operator.

TABLE 1 Period Trans. Ant. 1 (Tx₁) Trans Ant. 2 (Tx₂) Receive Ant. (Rx)T₁ s₁ s₂ r₁ T₂ −s₂*  s₁* r₂

The baseband signals during a first interval can be written as:r ₁ =h ₁ s ₁ +h ₂ s ₂ +n ₁  (1)r ₂ =−h ₂ s ₂ *+h ₁ s ₁ *+n ₂*  (2)

-   -   where n₁ and n₂ are noise factors and let us assume impulse        response coefficients, e.g. flat-fading case h_(i), where i=1 to        K associated with the two antennas (K=2) are constant over the        two-symbol time interval.

We can map the table into matrix form with the columns correspond toantennas and the row to time epochs, where the column index representsor is associates with the antenna index and the row index represents oris associated with the time index. Thus, the Alamouti Space-Time Blockcode (STBC) $\begin{matrix}{C_{Ala} = {\begin{matrix}T \\i \\m \\e \\ \downarrow \end{matrix}\overset{{Antenna}\quad->}{\begin{bmatrix}s_{1} & s_{2} \\{- s_{2}^{*}} & s_{1}^{*}\end{bmatrix}}}} & (3)\end{matrix}$is optimal with complex signal constellations. It reaches diversity 2,with a linear decoding scheme which yields estimates for both symbolswith the two channels maximal ratio combined.

The impulse response coefficients can be represented by$h = {\left. h \right\rangle = \begin{bmatrix}h_{1} \\h_{2}\end{bmatrix}}$and the received signals by$r = {\left. r \right\rangle = \begin{bmatrix}r_{1} \\r_{2}\end{bmatrix}}$and the noise by $n = {\left. n \right\rangle = {\begin{bmatrix}n_{1} \\n_{2}\end{bmatrix}.}}$

The signal received at Rx 130 is given by: $\begin{matrix}{\begin{bmatrix}r_{1} \\r_{2}\end{bmatrix} = {\left. {{C_{Ala}\begin{bmatrix}h_{1} \\h_{2}\end{bmatrix}} + n}\Leftrightarrow\left. r \right\rangle \right. = {\left\langle {h_{i}{C}h_{j}} \right\rangle + \left. n \right\rangle}}} & (4)\end{matrix}$

-   -   or, equivalently, $\begin{bmatrix}        r_{1} \\        r_{2}^{*}        \end{bmatrix} = {\left. {{\begin{bmatrix}        h_{1} & h_{2} \\        h_{2}^{*} & {- h_{1}^{*}}        \end{bmatrix}\quad\begin{bmatrix}        s_{1} \\        s_{2}        \end{bmatrix}} + \begin{bmatrix}        n_{1} \\        n_{2}^{*}        \end{bmatrix}}\Leftrightarrow r \right. = {{H_{12}s} + n}}$

There is interest to derive space-time codes (STC) for more than twoantennas. However, extension of the Alamouti method to more than twoantennas is not straightforward. Note that the channel matrix H₁₂ inequation (5) is orthogonal, thus to decode we use:s=H ₁₂ ^(H) r  (6)

The detection is:ŝ=sign (H ₁₂ ^(H) r)=sign[(||h ₁||² +||h ₂||²)I ₂ s+H ₁₂ ^(H) n]  (7)

The C_(Ala) matrix may be recognized as proportional to a generalunitary unimodular matrix which is commonly written in the art as:$\begin{matrix}{{U\left( {a,b} \right)} = \begin{bmatrix}a & b \\{- b^{*}} & a^{*}\end{bmatrix}} & (8)\end{matrix}$where a and b are complex numbers which satisfy the unimodular condition|a|²+|b|²=1. The orthogonality of the space-time block codes may thus beexpressed as $\begin{matrix}{{C^{H}C} = {\sum\limits_{i}{{z_{i}}^{2}I}}} & (9)\end{matrix}$

C is the code matrix, I is the identity matrix of the same dimension,and the superscript H is the hermitean operator (complex conjugatetranspose). The maximum-ratio-combining property of the code is a directconsequence of the appearance of the sum of the symbol powers on themain diagonal of the hermitean square of the code matrix.

In general, the Alamounti STBC is the Radon-Hurwitz submatrix form andis an unitary unimodular matrix given in general form by:$\begin{matrix}{C_{R\quad H} = \begin{bmatrix}c_{n} & c_{n + 1} \\{- c_{n + 1}^{*}} & c_{n}^{*}\end{bmatrix}} & (10)\end{matrix}$

The mathematical work of Radon and Hurwitz in the 1920's [proved that anorthogonal design of size N exists if and only if N=2, 4, and 8.] iscited in Calderbank et al. in U.S. Pat. No. 6,088,408 issued toCalderbank et al. on Jul. 11, 2000 and in (V. Tarokh, H. Jafarkhani, andA. Calderbank, “Space-Time Block Codes from Orthogonal Designs,” IEEETransactions on Information Theory, pp. 1456-1467, July 1999), bothincorporated herein by reference, showed the Hurwitz-Radon proof andshowed that for more than two antennas complex orthogonal designs thatachieve R=1 do not exist. Space-Time Block Codes. Calderbank et al.proposed a method using rate=½, and ¾ Space-Time Block codes fortransmitting on three and four antennas using complex signalconstellations.

As an example, the code rate ¾ is given by: $\begin{matrix}{{C_{\frac{3}{4}}\left( \quad{s_{1},\quad s_{2},\quad s_{3}} \right)} = {\quad\left\lbrack \quad\begin{matrix}s_{1} & s_{2} & \frac{s_{3}}{\sqrt{2}} & \frac{s_{3}}{\sqrt{2}} \\{- s_{2}^{*}} & s_{1}^{*} & \frac{s_{3}}{\sqrt{2}} & {- \frac{s_{3}}{\sqrt{2}}} \\\frac{s_{3}^{*}}{\sqrt{2}} & \frac{s_{3}^{*}}{\sqrt{2}} & \frac{\left( {s_{2} - s_{2}^{*} - s_{1} - s_{1}^{*}} \right)}{2} & \frac{\left( {s_{1} - s_{1}^{*} - s_{2} - s_{2}^{*}} \right)}{2} \\\frac{s_{3}}{\sqrt{2}} & {- \frac{s_{3}^{*}}{\sqrt{2}}} & \frac{\left( {s_{1} - s_{1}^{*} + s_{2} + s_{2}^{*}} \right)}{2} & \frac{\left( {s_{1} + s_{1}^{*} + {s2} - s_{2}^{*}} \right)}{2}\end{matrix}\quad \right\rbrack}} & (11)\end{matrix}$

This method has a disadvantage in a loss in transmission rate and thefact that the multi-level nature of the ST coded symbols increases thepeak-to-average ratio requirement of the transmitted signal and imposesstringent requirements on the linear power amplifier design. Othermethods proposed include a rate (R=1), orthogonal transmit diversity(OTD)+space-time transmit diversity scheme (STTD) four antenna method.

When complex modulation is used, full diversity codes with a code rate 1are only described in connection with two antennas in the publication WO99/14871 Published on Mar. 25, 1999 and entitled TRANSMITTER DIVERSITYTECHNIQUE FOR WIRELESS COMMUNICATIONS and the publication Tarokh, V.,Jafarkhani, H., Calderbank, A. R.: Space-Time Block Coding for WirelessCommunications: Performance Results, IEEE Journal on Selected Areas InCommunication, Vol. 17 pp. 451-460, March 1999, both incorporated hereinby reference present a rate ½ code which is constructed from the fullrate real code by setting the complex signals on top of the same, butconjugated signals. This way rate ½ codes for two to eight antennas areobtained. In the following, an example of a code for three antennas isgiven: $\begin{matrix}{{C_{1/2}\left( {s_{1},s_{2},s_{3},s_{4}} \right)}->\begin{bmatrix}s_{1} & s_{2} & s_{3} \\{- s_{2}} & s_{1} & {- s_{4}} \\{- s_{3}} & s_{4} & s_{1} \\{- s_{4}} & {- s_{3}} & s_{2} \\s_{1}^{*} & s_{2}^{*} & s_{3}^{*} \\{- s_{2}^{*}} & s_{1}^{*} & {- s_{1}^{*}} \\{- s_{3}^{*}} & s_{4}^{*} & s_{1}^{*} \\{- s_{4}^{*}} & {- s_{3}^{*}} & s_{2}^{*}\end{bmatrix}} & (12)\end{matrix}$

-   -   where a star (*) refers to a complex conjugate. These codes are        not delay-optimal.

So far, all complex space-time block codes have belonged to twocategories: a group based on real codes, halving the code rate, such asthe above example, or a group based on square unitary matrices.

It is desirable that ‘Open-loop diversity’ should have these fourproperties:

-   -   1. Full diversity in regard to the number of antennas.    -   2. Only linear processing is required in a transmitter and a        receiver.    -   3. Transmission power is divided equally between the antennas.    -   4. The code rate efficiency is as high as possible.

A drawback of the above solutions is that only the requirements 1 and 2can be fulfilled. For example, the transmission power of differentantennas is divided unequally, (i.e. different antennas transmit atdifferent powers). This causes problems in the planning of outputamplifiers. Furthermore, the code rate is not optimal.

For 3 and 4 antennas, this maximal rate is ¾. Because of theinefficiency of codes with rates less than one (R<1) transmit power of agiven antenna fluctuates in time; thus, presenting a power-unbalancedproblem. Therefore, there is a need to provide a power-balance full-ratecode. Since a decrease in rate may not acceptable, some other featuresof space-time block codes have to be relaxed.

For example, uncoded diversity gain may be sacrificed and rely on codingto exploit the diversity provided by additional antennas. Motorolaintroduced Orthogonal Transmit Diversity+Space-Time Transmit Diversity(OTD+STTD) scheme, (L. Jalloul, K. Rohani, K. Kuchi, and J. Chen,“Performance Analysis of CDMA Transmit Diversity Methods,” Proceedingsof IEEE Vehicular Technology Conference, vol. 3, pp. 1326-1330 Fall1999, hereinafter referred to as Jalloul). Jalloul states that extensionof OTD to more than two antennas is straightforward; but “STTD is notdirectly extendable to more than two antennas, since rate one S-T blockcodes (STC) are non-existent for greater that 2 antennas.” They extendedthe two antenna STTD scheme to four transmit antennas by combining theSTTD with two-branch OTD, $\begin{matrix}{{C_{STOTD}\left( {s_{1},s_{2},s_{3},s_{4}} \right)} = \begin{bmatrix}s_{1} & s_{2} & s_{1} & s_{2} \\{- s_{2}^{*}} & s_{1}^{*} & {- s_{2}^{*}} & s_{1}^{*} \\s_{3} & s_{4} & s_{3} & s_{4} \\{- s_{4}^{*}} & s_{3}^{*} & {- s_{4}^{*}} & s_{3}^{*}\end{bmatrix}} & (13)\end{matrix}$

The STOTD scheme is completely balanced and is also orthogonal, so thatlinear decoding gives maximal likelihood results. However, the diversityorder achieved is only two, which is the same as the Alamouti STTDscheme.

Space-Time Code Design Criteria

There are three design criteria for space-time block codes, which areall formulated in terms of the codeword difference matrixD_(ce)=C_(c)−C_(e), where C_(c) and C_(e) are the code matricescorresponding to two distinct sets of information c and e. Minimizingthe pair-wise error probability of deciding in favor of C_(e) whentransmitting C_(c) leads to the following design criteria:

-   -   1. The rank criterion: The diversity gained by a multiple        transmitter scheme is:        diversity=min_(e≠c)Rank[D_(ce)]<=min[T;N]  (14)

To achieve maximal diversity, D_(ce) should have full rank for alldistinct code words c and e.

-   -   2. The determinant criterion: To optimize performance in a        Rayleigh fading environment, Code (C) should be designed to        maximize        min_(e≠s)det′[D_(ce) ^(H)D_(ce)].  (15)

Where the prime in the determinant indicates that zero eigenvaluesshould be left out from the product of eigenvalues when computing thedeterminant.

-   -   3. The trace criterion: To optimize performance in flat fading        channels, Code (C) should be designed to maximize the Euclidean        distance        Tr[D_(ce) ^(H)D_(ce)].  (16)

Moreover, to optimize performance in fading channels, the eigenvalues ofD_(ce) ^(H) D_(ce) should be as close to each other as possible.

From linearity, it follows that the codeword difference matrix D_(ce)inherits the unitarity property of the code matrix C: $\begin{matrix}{{D_{ce}^{H}D_{ce}} = {\sum\limits_{k}{{{z_{k,c} - z_{k,e}}}^{2}{I_{N}.}}}} & (17)\end{matrix}$

Thus, all design criteria are fulfilled:

-   -   Rank criterion: As an unitary matrix, D_(ce) is full rank for        all distinct code word pairs. Thus, all space-time block codes        give full diversity, equaling the number of Tx antennas.

Determinant criterion: As D_(ce) is unitary, $\begin{matrix}{{\det{{D_{ce}^{H}D_{ce}}}} = {\sum\limits_{k}{{z_{k,c} - z_{k,e}}}^{2N}}} & (18)\end{matrix}$

This is the maximum given a fixed transmit power.

Trace criterion: As D_(ce) is unitary,${{Tr}{{D_{ce}^{H}D_{ce}}}} = {N{\sum\limits_{k}{{{z_{k,c} - z_{k,e}}}^{2}.}}}$This is the maximum given a fixed transmit power. Moreover, alleigenvalues are the same.

Thus, there is a need for a design which provides full, diversity, fullrate and power-balanced space-time codes.

SUMMARY OF THE INVENTION

It is thus an object of the invention to implement a method and a systemby which optimal diversity is achieved with different numbers ofantennas. This is achieved by a method of transmitting a digital signalconsisting of symbols, which method comprises the steps of codingcomplex symbols to channel symbols in blocks having the length of agiven K and transmitting the channel symbols via several differentchannels and two or more antennas. In the method of the invention,coding is performed such that the coding is defined by a code matrix,which can be expressed as a sum of 2K elements, in which each element isa product of a symbol or symbol complex conjugate to be transmitted anda N×N representation matrix of a complexified anti-commutator algebra,extended by a unit element, and in which each matrix is used at mostonce in the formation of the code matrix.

Further, in the method of the invention the coding is performed suchthat the coding is defined by a code matrix which is formed by freelyselecting 2K−1 unitary, anti-hermitean N×N matrices anti-commuting witheach other, forming K−1 pairs of said matrices, whereby the remainingmatrix forms a pair with an N-dimensional unit matrix, forming twomatrices of each pair such that the second matrix of the pair,multiplied by the imaginary unit, is added to and subtracted from thefirst matrix of the pair, and in which each matrix formed in the abovemanner defines the dependence of the code matrix on one symbol or symbolcomplex conjugate to be coded.

The invention also relates to an arrangement for transmitting a digitalsignal consisting of symbols, which arrangement comprises a coder forcoding complex symbols to channel symbols in blocks having the length ofa given K, means for transmitting the channel symbols via severaldifferent channels and two or more antennas. In the arrangement of theinvention, the coder is arranged to code the symbols using a codematrix, which can be expressed as a sum of 2K elements, in which eachelement is a product of a symbol or symbol complex conjugate to betransmitted and a a N×N representation matrix of a complexifiedanti-commutator algebra, extended by a unit element, and in which eachmatrix is used at most once in the formation of the code matrix.

Furthermore, in the arrangement of the invention the coder is arrangedto code the symbols using a code matrix which is formed by freelyselecting 2K-1 unitary, anti-hermitean N×N matrices anti-commuting witheach other, forming K−1 pairs of said matrices, whereby the remainingmatrix forms a pair with an N-dimensional unit matrix, forming twomatrices of each pair such that the second matrix of the pair,multiplied by the imaginary unit, is added to and subtracted from thefirst matrix of the pair, and in which each matrix formed in the abovemanner defines the dependence of the code matrix on one symbol or symbolcomplex conjugate to be coded.

The solution of the invention can provide a system in which any numberof transmit and receive antennas can be used and a full diversity gaincan be achieved by space-time block coding. In a preferred embodiment,the maximal code rate and the optimal delay are achieved by square codeshaving a dimension that is a power of two.

The solution of the invention employs complex block codes. In apreferred embodiment codes are used, which are based on matrices whoseall elements have the form of ±s_(k), ±s*_(k) or 0. The prior artsolutions reveal no codes in whose elements the term 0 appears. First,square codes are given, from which non-square codes are obtained byeliminating columns (antennas). In these codes known as basic codes theelements depend only on one symbol, or on the real part of a symbol andthe imaginary part of another symbol. In another preferred embodiment,full diversity codes which do not have the above restriction can beused.

Also provided is a method and arrangement in which a code matrix isformed by matrices of a portion of the symbols placed on the diagonal ofthe code matrix and by matrices of a second portion of symbols along theanti-diagonal of the code matrix.

A sub-optimal solution is also provided by an embodiment of theinvention using a rate ⅓ convolution code.

An equal distribution of transmission power between different antennasis also achieved by means of the solution of the invention. The solutionof the invention preferably also provides coding in which the ratio ofthe maximum power to the average power or the ratio of the average powerto the minimum power can be minimized.

At the receiver, the transmitted symbols may be recovered using amaximum likelihood sequence estimator (MLSE) decoder implemented withthe Viterbi algorithm with a decoding trellis according to thetransmitter.

A BRIEF DESCRIPTION OF THE DRAWINGS

The above set forth and other features of the invention are made moreapparent in the ensuing Detailed Description of the Invention when readin conjunction with the attached Drawings, wherein:

FIG. 1 shows an example of a system of multi-channel transmission inwhich multi-path path fading may occur;

FIG. 2A shows an example of a system in accordance with an embodiment ofthe invention;

FIG. 2B shows another example of a system in accordance with anembodiment of the invention;

FIG. 3 is an illustration showing an example of a transmitter/receiverarrangement in accordance with an embodiment of the invention;

FIG. 4 shows a rate {fraction (5/16)} two-layer block code for fourantennas;

FIG. 5 shows a rate {fraction (63/64)} three-layer block code for fourantennas; and

FIGS. 6A & 6B show the two forms of matrices from which a power-balancedfull rate code may be constructed.

DETAILED DESCRIPTION

The invention may be used in radio systems which allow the transmissionof at least a part of a signal by using at least three or more transmitantennas or three or more beams that are accomplished by any number oftransmit antennas. A transmission channel may be formed by using a timedivision, frequency division or code division multiple access method.Also systems that employ combinations of different multiple accessmethods are in accordance with the invention. The examples describe theuse of the invention in a universal mobile communication systemutilizing a broadband code division multiple access method implementedwith a direct sequential technique, yet without restricting theinvention thereto.

Referring to FIG. 2A, a structure of a mobile communication system isdescribed by way of example. The main parts of the mobile communicationsystem are core network CN, UMTS terrestrial radio access network UTRANand user equipment UE. The interface between the CN and the UTRAN iscalled lu and the air interface between the UTRAN and the UE is calledUu.

The UTRAN comprises radio network subsystems RNS. The interface betweenthe RNSs is called lur. The RNS comprises a radio network controller RNCand one or more nodes B. The interface between the RNC and B is calledlub. The coverage area, or cell, of the node B is marked with C in thefigure.

The description of FIG. 2A is relatively general, and it is clarifiedwith a more specific example of a cellular radio system in FIG. 2B. FIG.2B includes only the most essential blocks, but it is obvious to aperson skilled in the art that the conventional cellular radio systemalso includes other functions and structures, which need not be furtherexplained herein. It is also to be noted that FIG. 2B only shows oneexemplified structure. In systems according to the invention, detailscan be different from what is shown in FIG. 2B, but as to the invention,these differences are not relevant.

A cellular radio network typically comprises a fixed networkinfrastructure, i.e. a network part 200, and user equipment 202, whichmay be fixedly located, vehicle-mounted or portable terminals. Thenetwork part 200 comprises base stations 204, a base stationcorresponding to a B-node shown in the previous figure. A plurality ofbase stations 204 are, in turn, controlled in a centralized manner by aradio network controller 206 communicating with them. The base station204 comprises transceivers 208 and a multiplexer 212.

The base station 204 further comprises a control unit 210 which controlsthe operation of the transceivers 208 and the multiplexer 212. Themultiplexer 212 arranges the traffic and control channels used byseveral transceivers 208 to a single transmission connection 214, whichforms an interface lub.

The transceivers 208 of the base station 204 are connected to an antennaunit 218 which is used for implementing a bi-directional radioconnection 216 to the user equipment 202. The structure of the frames tobe transmitted in the bi-directional radio connection 216 is definedseparately in each system, the connection being referred to as an airinterface Uu.

The radio network controller 206 comprises a group switching field 220and a control unit 222. The group switching field 220 is used forconnecting speech and data and for combining signalling circuits. Thebase station 204 and the radio network controller 206 form a radionetwork subsystem 224 which further comprises a transcoder 226. Thetranscoder 226 is usually located as close to a mobile servicesswitching center 228 as possible, because speech can then be transferredin a cellular radio network form between the transcoder 226 and theradio network controller 206, which saves transmission capacity.

The transcoder 226 converts different digital speech coding forms usedbetween a public switched telephone network and a radio network to makethem compatible, for instance from a fixed network form to anothercellular radio network form, and vice versa. The control unit 222performs call control, mobility management, collection of statisticaldata and signalling.

FIG. 2B further shows the mobile services switching center 228 and agateway mobile services switching center 230 which controls theconnections from the mobile communications system to the outside world,in this case to a public switched telephone network 232.

In accordance with the present invention there is provided a sub-optimalclass of Space-Time Codes (STC) based on the Radon-Hurwitz sub-matrix.The invention can thus be applied particularly to a system in whichsignal transmission is carried out by using complex space-time blockcoding in which the complex symbols to be transmitted are coded tochannel symbols in blocks having the length of a given K in order to betransmitted via several different channels and two or more antennas.These several different channels can be formed of different time slots.As a result of the coding, the symbol block forms into a code matrix inwhich the number of columns corresponds to the number of antennas usedfor the transmission and the number of rows corresponds to the number ofdifferent channels, which, in case of space-time coding, is the numberof time slots to be used. Correspondingly, the invention can be appliedto a system in which different frequencies or different spreading codesare used instead of time slots. In this case it does not naturally dealwith space-time coding but rather with space-frequency coding orspace-code-division coding. The space-frequency coding could be used inan OFDM (orthogonal frequency division multiplexing) system, forexample.

The codes of the present invention have rate R=1 and are easilydecodable. For the 4-antenna case, a diversity of order 3 is achieved.The present invention minimizes the inherent non-orthogonality and havesimple linear decoding, which may be iterated. The present invention isalso backward compatible with 3GPP release 99 open-loop diversity mode.

Let us first examine the forming of a freely selected square complexspace-time block code. Assuming the number of transmit antennas isN=2^(K−1), where K is an integer and bigger than two. By means of theobtained code, K complex number modulated symbols can be transmittedduring N symbol periods. These symbols can be marked with s_(k), k=1, .. . K.

A square complex space-time block code is based on a unitary N×N matrix,whose elements depend on linearly transmitted symbols s_(k) and theircomplex conjugates. A unitary matrix is a square matrix whose inversematrix is proportional to its hermitean conjugate. On the other hand,the hermitean conjugate is the complex conjugate of the matrixtranspose. In addition, the proportional coefficient between the productof the code matrix and its hermitean conjugate, and the unit matrix is alinear combination of the absolute value squares of the symbols to betransmitted. This linear combination can be called unitaritycoefficient. By interpreting the symbols to be transmittedappropriately, this linear combination can always be seen as a sum ofthe absolute value squares of the symbols to be transmitted.

The unitarity (complex-orthogonality) of the space-time block codes maythus be expressed with the inner products proportional to the sum of thesquared amplitudes of the symbols: $\begin{matrix}{{C^{H}C} = {\sum\limits_{i}{{z_{i}}^{2}I}}} & (19)\end{matrix}$

C is the code matrix, I is the identity matrix of the same dimension,and H is the hermitean conjugate (complex conjugate transpose operator).The maximum-ratio-combining property of the code is a direct consequenceof the appearance of the sum of the symbol powers on the diagonal of thehermitean square of the code matrix.

The linearity of the symbols z₁, z₂ allows the code matrix to beexpanded thus $\begin{matrix}{C = {\sum\limits_{k = 1}^{K}\left( {{z_{k}\beta_{k}^{-}} + {z_{k}^{*}\beta_{k}^{+}}} \right)}} & (20) \\{= {\sum\limits_{k = 1}^{K}{\left( {{x_{k}\beta_{{2k} - 2}} + {y_{k}\beta_{{2k} - 1}}} \right).}}} & (21)\end{matrix}$

{β}_(k=0) ^(2K−1) is a set of 2K constant T×N matrices with complexentries. The real variable x_(k) and y_(k) are the real and imaginarypart of the complex variable z_(k)=x_(k)+jy_(k), and the matrices β_(k)^(±) are linear combination of β_(k) $\begin{matrix}{\beta_{k}^{\pm} = {\frac{1}{2}{\left( {\beta_{{2k} - 2} \pm {j\quad\beta_{{2k} - 1}}} \right).}}} & (22)\end{matrix}$

Combining the above based on unitarity and linearity criteria thefollowing is a restriction on the β_(k) matrices.β_(k) ^(H)β_(j)+β_(j) ^(H)β_(k)=2δ_(jk) I _(N).  (23)

For square matrices (T=N), it can be seen the code matrix C(z) and thecoefficient matrices β_(k) are unitary which lends itself to the use ofgroup algebras. Moreover, the matrices belong to the unitary group U(N).

As stated above in regards to Pauli-Spin matrices, taking a set oflinearly independent vectors as group elements, a vector space may begenerated by taking a linear combination of the elements of the set. Theproduct already defined on the basis set can be extended to all theelements of the vector space by linearity. Thus, defining a groupalgebra. Due to this, the well developed tools of the representationtheory of continuous groups lend themselves to the analysis of theproblem.

Redefineγ_(k)=β_(k) ^(H)β_(k) ;k=0, . . . ,2K−1,  (24)

γ₀=I_(N), and the algebra β_(k) ^(H)β_(j)+β_(j) ^(H)β_(k)=2δ_(jk)I_(N)from above remains unchanged for the γ:s. From the relation between γ₀and the other γ:s, we then see that these should be anti-hermiteanγ_(k) ^(H)=−γ_(k) ,k=1, . . . ,2K−1,  (25)

The algebra of these remaining γ:s is nowγ_(k)γ_(j)+γ_(j)γ_(k)=−2δ_(jk) I _(N) ,j,k=1, . . . ,2K−1.  (26)γ_(k)γ_(j)

This is the defining relation of generators of the Clifford algebra.There is a coincidence that the investigation of the Clifford algebraoriginates in the work of theoretical physicist P.A.M. Dirac onfermionic particles in space-time. From a group theoretical point ofview, the γ-matrices generate the so-called spinor representation of thespecial orthogonal subgroup SO(2K−1), and here we are dealing with theproblem of embedding spinorial SO(2K−1) representation inrepresentations of unitary groups.

We thus get the following generic prescription for finding a complexmodulation space-time block code:

-   -   1. Find a N-dimensional representation of the Clifford algebra        γ_(k)γ_(j)+γ_(j)γ_(k)=−2δ_(jk)I_(N),j,k=1 . . . ,2K −1 for        anti-hermitean matrices γ_(k), k=1, . . . ,2K−1.    -   2. Take an unitary matrix β₀εU(N).    -   3. Define β_(k)=β₀γ_(k),k=1, . . . ,2K−1.    -   4. Use the matrices β_(k),k=0, . . . ,2K−1 to create a code        matrix C(z) according        ${{to}\quad C} = {\sum\limits_{k = 1}^{K}{\left( {{x_{k}\beta_{{2k} - 2}} + {y_{k}\beta_{{2k} - 1}}} \right).}}$

By construction, this prescription yields all possible complex linearspace-time codes with N antennas and time epochs, full diversity, andrate K/N. The codes with minimal dimensions for a given rate are thusdelay optimal codes.

Similar to what was shown as a result of the Hurwitz-Radon proof, therepresentation theory of Clifford algebas give very stringent conditionson the existence of block codes with arbitrary rate K/N. For anorthogonal complex modulation space-time block code employing Nantennas, the maximal achievable rate is: $\begin{matrix}\frac{\left\lbrack {\log_{2}N} \right\rbrack + 1}{2^{\lbrack{\log_{2}N}\rbrack}} & (27)\end{matrix}$

However, one may not always need optimal codes. An embodiment of theinvention provides a sub-optimal solution as will be presented below.

Representation Theory of Clifford Algebras

Let us look at an example using Clifford algebra representation. Supposeyou have K objects that fulfil the defining relations of the generatorsof a Clifford algebra γ_(k)γ_(j)+γ_(j)γ_(k)=−2δ_(jk)I_(N),j,k=1, . . .,2K−1.That is, they are fourth roots of the identity matrix, (i.e. gεGg²=−I), and they anti-commute. We want to represent these objects asanti-hermitean square matrices, and we are interested in the minimaldimension of irreducible representations. (An irreducible representationis one that cannot be decomposed into a direct product of twolower-dimensional ones). For anti-hermitean matrices, idempotencytranslates to unitarity. The enveloping algebra of these objects is thealgebra Clifford_(K). We shall use this terminology loosely, and referwith “Clifford_(K)” to the algebra of the generators of Clifford_(K).

In constructing representations for Clifford algebras, we shall use thefollowing matrices: $\begin{matrix}{{\sigma_{1} = \begin{pmatrix}0 & 1 \\{- 1} & 0\end{pmatrix}},{\sigma_{2} = \begin{pmatrix}0 & I \\I & 0\end{pmatrix}},{{{and}\quad\sigma_{3}} = \begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}}} & (28)\end{matrix}$

Matrices (28) are proportional to Pauli matrices. For later convenience,these have been chosen so that σ₁ and or σ₂ are anti-hermitean, and σ₃is hermitean. The matrices that σ₁, σ₂, and I σ₃ constitute a basis forthe algebra Su₂, the algebra of anti-hermitean 2×2 matrices withvanishing trace, which is a real form of Sl₂(C)=A1 in Cartan'sclassification of Lie algebras. The fundamental representation (theminimal dimension faithful representation), is generated by these 2×2matrices.

Minimal Dimensions for Irreducible Representations

First consider the case K=2. Clearly, the objects γ₁, γ₂ cannot berepresented as complex numbers. The minimal dimension to represent twoanti-commuting objects is 2. It is easy to find two anti-commutinganti-hermitean unitary 2×2 matrices, take e.g.γ₁=σ₁; γ₂=σ₂.  (29)

Now consider a Clifford-algebra generated by three elements γ₁;γ₂; γ₃.From the anti-commutation relations, it follows that the product γ₁γ₂;γ₃commutes with all generators of the algebra. As an hermitean/Casimiroperator, its matrix representation can be taken to be proportional tothe unit matrix I₂. Thus, γ₃ can be represented in the same matrixdimension as γ₁ and γ₂, and is simply proportional to their product.Corresponding to γ₁,=σ₁; γ₂=σ₂. we have e.g.γ₃=γ₁ γ₂=Iσ₃.  (30)Here we see that Clifford₃ coincides with the algebra su₂. Also,Clifford₂ is a two-dimensional sub-algebra of su₂.

Taking the matrices β in$C = {\sum\limits_{k = 1}^{K}{\left( {{x_{k}\beta_{{2k} - 2}} + {y_{k}\beta_{{2k} - 1}}} \right).}}$to be the above representation of the three generators of Clifford₃,together with the two-dimensional unit matrix, we get exactly the 2×2Alamouti code. Next add a fourth generator to the Clifford-algebra.Denoting {tilde over (γ)}₃=−jγ₁γ₂γ₃, {tilde over (γ)}₄=−jγ₁γ₂γ₄, we seefrom the anti-commutation relations that the sub-algebras generated byγ₁, γ₂ and {tilde over (γ)}₃, {tilde over (γ)}₄ commute. The twosub-algebras, on the other hand, are isomorphic to Clifford₂. Thus eachof these subalgebras can be represented using the matrices in Equation(A.27). The two commuting sub-algebras, however, have to be representedusing the tensor product. Thus, we get the following matrixrepresentation for the generators of Clifford₄:γ₁ =I ₂{circle around (×)}σ₁;γ2=I ₂{circle around (×)}σ₂ {tilde over(γ)}₃=σ₁{circle around (×)}I ₂;{tilde over (γ)}₄=σ₂{circle around (×)}I₂.  (31)

Adding a fifth generator, we again note that jγ₁ γ₂ γ₃ γ₄ γ₅ is anhermitean/Casimir operator, and its matrix representation can be takenproportional to the unit matrix. Thus, we can define e.g. γ₅=−jγ₁γ₂γ₃γ₄.

We have constructed the following representation of Clifford₅:$\begin{matrix}{{\gamma_{1} = \begin{bmatrix}0 & 1 & 0 & 0 \\{- 1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & {- 1} & 0\end{bmatrix}},{\gamma_{2} = \begin{bmatrix}0 & j & 0 & 0 \\j & 0 & 0 & 0 \\0 & 0 & 0 & j \\0 & 0 & j & 0\end{bmatrix}},{\gamma_{3} = \begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & {- 1} \\{- 1} & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{bmatrix}},{\gamma_{4} = \begin{bmatrix}0 & 0 & j & 0 \\0 & 0 & 0 & {- j} \\j & 0 & 0 & 0 \\0 & {- j} & 0 & 0\end{bmatrix}},{\gamma_{5} = \begin{bmatrix}j & 0 & 0 & 0 \\0 & {- j} & 0 & 0 \\0 & 0 & {- j} & 0 \\0 & 0 & 0 & j\end{bmatrix}},} & (32)\end{matrix}$

To get the form $\begin{matrix}{{C_{3/4}\left( {s_{1},s_{2},s_{3}} \right)} = \begin{bmatrix}s_{1} & s_{2} & s_{3} & 0 \\{- s_{2}^{*}} & s_{1}^{*} & 0 & {- s_{3}} \\{- s_{3}^{*}} & 0 & s_{1}^{*} & s_{2} \\0 & s_{3}^{*} & {- s_{2}^{*}} & s_{1}\end{bmatrix}} & (33)\end{matrix}$

-   -   using prescription        ${C = {\sum\limits_{k = 1}^{K}{\left( {{x_{k}\beta_{{2k} - 2}} + {y_{k}\beta_{{2k} - 1}}} \right).}}},$        one can use the matrices above with the indices cyclically        permuted.

The same construction can be applied to find inductively the irreduciblerepresentations of a Clifford algebra generated by an arbitrary numberof elements K. There are two different sets of induction steps, fromeven K−1 to odd K, and from even K−2 to even K. For even K, theinduction steps are the following:{tilde over (γ)}_(K)=(−j)^(K/2−1)(Π_(i=1) ^(K−2) γ _(j))γ_(K),

Define{tilde over (γ)}_(K−1)=(−j)^(K/2−1)(Π_(i=1) ^(K−2)γ_(j))γ_(K−1).  (34)

Note that {γ_(j)}_(j=1) ^(K−2) and {{tilde over (γ)}_(K−1),{tilde over(γ)}_(K)}generate commuting Clifford sub-algebras isomorphic toClifford_(K−2) and Clifford₂.

Use a tensor product of existing irreducible representations R_(K−2) ofClifford_(K−2) and R₂ of Clifford₂ to represent these sub-algebras:R _(K)(γ_(j))=I ₂ {circle around (×)}R _(K−2)(γ_(j))j=1, . . . K−2R _(K)({tilde over (γ)}_(K−1))=R ₂(γ₁){circle around (×)}I _(dim) R_(K−2)R _(K)({tilde over (γ)}_(K))=R ₂(γ₂){circle around (×)}I _(dim) R_(K−2)  (35)

Finally, invert the definitions of the {tilde over (γ)}:s,$\begin{matrix}{{R_{K}\left( \gamma_{\{{{K - 1},K}\}} \right)} = {j^{{K/2} - 1}{R_{K}\left( {\prod\limits_{j = 1}^{K - 2}\quad{\gamma_{j}{\overset{\sim}{\gamma}}_{\{{{K - 1},K}\}}}} \right)}}} & (36)\end{matrix}$

The induction steps from even K−1 to odd K are following $\begin{matrix}{{{Define}\quad\Lambda_{K}} = {\left( {- j} \right)^{{({K + 1})}/2}{\prod\limits_{j = 1}^{K}\quad\gamma_{j}}}} & (37)\end{matrix}$

Note that Λ_(K) is a Casimir operator; it commutes with all{γ_(j)}_(j=1) ^(K).

Represent it by the unit operator in the representation R_(K−1).

Take

R _(K)(γ_(j))=R _(K−1)(γ_(j)),j=1, . . . K−1  (38) $\begin{matrix}{{R_{K}\left( \gamma_{K} \right)} = {j^{{({K + 1})}/2}{R_{K - 1}\left( {\prod\limits_{j = 1}^{K - 1}\quad\gamma_{j}} \right)}}} & (39)\end{matrix}$

The induction steps above can be used to prove the following theorem:The minimal dimension to represent a Clifford algebra generated by Kelements is 2^([K/2]). The minimal representation matrices are elementsin the space $\begin{matrix}{\underset{\underset{{\lbrack{K/2}\rbrack}{elements}}{︸}}{R_{K}^{minimal} = {{{\overset{\sim}{R}}_{2}\left( {su}_{2} \right)} \otimes {{\overset{\sim}{R}}_{2}\left( {su}_{2} \right)} \otimes \ldots \otimes {{\overset{\sim}{R}}_{2}\left( {su}_{2} \right)}}}.} & (40)\end{matrix}$

Here, R₂(su₂) is the fundamental two-dimensional representation of su₂,the algebra of anti-hermitean 2×2 matrices with vanishing trace. {tildeover (R)} denotes this representation space enlarged by matricesproportional to the two-dimensional unit matrix.

A minimal (i.e. 2^(K−1)) dimensional representation of Clifford_(K)constructed by the induction described above, is $\begin{matrix}\begin{matrix}{\gamma_{2} = {\underset{\underset{K - {2\quad{times}}}{︸}}{I_{2} \otimes I_{2} \otimes \ldots \otimes I_{2} \otimes I_{2}} \otimes \sigma_{1}}} \\{\gamma_{3} = {\underset{\underset{K - {3\quad{times}}}{︸}}{I_{2} \otimes I_{2} \otimes \ldots \otimes I_{2} \otimes I_{2}} \otimes \sigma_{2}}} \\{\gamma_{4} = {\underset{\underset{K - {3\quad{times}}}{︸}}{I_{2} \otimes I_{2} \otimes \ldots \otimes I_{2} \otimes I_{2}} \otimes \sigma_{1} \otimes \sigma_{3}}} \\{\gamma_{5} = {\underset{\underset{K - {3\quad{times}}}{︸}}{I_{2} \otimes I_{2} \otimes \ldots \otimes I_{2} \otimes I_{2}} \otimes \sigma_{2} \otimes \sigma_{3}}} \\{\gamma_{6} = {\underset{\underset{K - {4\quad{times}}}{︸}}{I_{2} \otimes \ldots \otimes I_{2}} \otimes \sigma_{1} \otimes \sigma_{3} \otimes \sigma_{3}}} \\{\gamma_{7} = {\underset{\underset{K - {4\quad{times}}}{︸}}{I_{2} \otimes \ldots \otimes I_{2}} \otimes \sigma_{2} \otimes \sigma_{3} \otimes \sigma_{3}}} \\{\gamma_{2k} = {\underset{\underset{K - 1 - {k\quad{times}}}{︸}}{I_{2} \otimes \ldots \otimes I_{2}} \otimes \sigma_{1} \otimes \underset{\underset{K - {1\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}} \\{\gamma_{{2k} + 1} = {\underset{\underset{K - 1 - {k\quad{times}}}{︸}}{I_{2} \otimes \ldots \otimes I_{2}} \otimes \sigma_{2} \otimes \underset{\underset{K - {1\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}} \\{\vdots} \\{\gamma_{{2K} - 2} = {\sigma_{1} \otimes \underset{\underset{K - {2\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}} \\{\gamma_{{2K} - 1} = {\sigma_{2} \otimes \underset{\underset{K - {2\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}} \\{\gamma_{1} = {j\underset{\underset{K - {1\quad{times}}}{︸}}{\sigma_{3} \otimes \sigma_{3} \otimes \ldots \otimes \sigma_{3}}}}\end{matrix} & (41)\end{matrix}$

To classify all possible representations, one has to assess the freeparameters in this construction.

First, note that the anti-commutation relationsγ_(k)γ_(j)+γ_(j)γ_(k)=−2δ_(jk)I_(N),j,k=1, . . . ,2K−1 have the symmetryγ_(j) →V ^(H)γ_(j) V,j=1 . . . ,L  (42)where V is an unitary dimR_(K)×dimR_(K) matrix. This symmetry is largeenough to accommodate any choice of basis in Clifford₂ sub-algebras.

Second, it is easy to see that representation in higher than the minimaldimensions can be found by taking any three higher dimensionalanti-commuting matrices instead of σ₁, σ₂, and σ₃ as a basis in any ofthe elements in the tensor-product above. Anti-commuting fourth roots ofthe identify matrix, however, are hard to come by.

A generic representation space would thus be $\begin{matrix}{R_{K}^{({{d1},\ldots\quad,d_{\lbrack{K/2}\rbrack}})} = {\left( {\oplus_{k = 1}^{d_{1}/2}{{\overset{\sim}{R}}_{2}\left( {su}_{2} \right)}} \right) \otimes \left( {\oplus_{k = 1}^{d_{2}/2}{{\overset{\sim}{R}}_{2}\left( {su}_{2} \right)}} \right) \otimes \ldots \otimes \left( {\oplus_{k = 1}^{d_{\lbrack{K/2}\rbrack}/2}{{\overset{\sim}{R}}_{2}\left( {su}_{2} \right)}} \right)}} & (43)\end{matrix}$

-   -   where the dimensions of the commuting sub-algebra representation        (d₁, . . . ,d_([K/2])) are even integers ≧2.

The above can be placed in a less rigorous mathematical narrative. Bymultiplying by a unitary N×N matrix, a square space-time block codewhich is freely selected from the left can be brought to a form in whichthe real part of a symbol to be transmitted appears only on the diagonalof the code matrix. If the symbols to be transmitted are interpreted inthe above manner, said real part appears in every diagonal element,multiplied by the same real number. In this case, the dependence of thecode matrix on the real part of the symbol is proportional to anN-dimensional unit matrix.

Let us next examine a method in which a unitary N×N matrix is formed,the elements of which depend linearly on symbols s_(k), the unitaritycoefficient of which is proportional to the sum of the absolute valuesquares of the symbols s_(k) and the dependence of which on the realpart of a symbol so is proportional to an N-dimensional unit matrix.

Let us take a freely selected 2K−1 quantity of N×N matrices, which areall antihermitean and unitary and which all anti-commute with eachother. An anti-hermitean matrix refers to a matrix, the hermiteanconjugate of which is the matrix itself multiplied by −1.Anti-commutation means that when two matrices can be multiplied by eachother in two orders, then if one product is −1 times the other product,the matrices anti-commute. The above family, to which 2K−1 matricesbelong, can be called an N-dimensional anti-commutator algebrapresentation of 2K−1 elements.

Let us form K−1 pairs of these 2K−1 matrices. Since there is an unevennumber of matrices, an N-dimensional unit matrix is used to form a pairwith the remaining matrix. Two matrices are formed of each matrix pairsuch that the second matrix of the pair, multiplied by the imaginaryunit, is added to and subtracted from the first matrix of the pair. Theunit matrix is interpreted in its own pair as the first matrix. Thisway, 2K matrices are formed. These matrices form a complexifiedanti-commutator algebra extended by a unit element. In short they arecalled complex anti-commutator matrices.

A code matrix is formed such that each of the matrices formed as abovedefines the dependence of the code matrix on one and only one s_(k) orthe complex conjugate of s_(k). Thus, the code matrix is the sum of 2Kelements and each element is the product of some s_(k) or s_(k) complexconjugate and an N×N complex anti-commutator matrix, such that eachsymbol, complex conjugate, and matrix only appears once in theexpression.

Let us examine a method of forming an N-dimensional anti-commutatoralgebra presentation of 2K−1 elements.

First, three 2×2 matrices are freely selected, the matrices fulfillingthe following conditions:

-   -   matrices are anti-hermitean and unitary; and    -   matrices anti-commute with each other.

So, the matrices form an anti-commutator algebra presentation of thefreely selected 3 elements. Two matrices are selected from the abovedefined matrices, and they can be called an elementary pair. Theremaining matrix is multiplied by the imaginary unit, and the result iscalled a third elementary matrix. In addition, a matrix proportional toa two-dimensional unit matrix is used as a fourth elementary matrix.This matrix can be called an elementary unit matrix.

K-I pairs of N×N matrices are formed of these matrices by formulatingtensor products of K−1 elementary matrices for example in the followingmanner:

-   -   The first matrix pair is established as a tensor product of K−2        elementary unit matrices and members of the elementary pair.        Each member of the elementary pair appears as separately        tensored with the unit matrices. This gives two matrices, (i.e.        a matrix pair).

The second matrix pair is obtained by tensoring K−3 elementary matrices,one member of the elementary pair and the third elementary matrix, inthis order.

The Ith matrix pair is obtained by tensoring K-I−1 elementary unitmatrices, one member of the elementary pair and I−1 third elementarymatrices, in this order.

K-1th pair is obtained by tensoring one member of the elementary pairand K−2 third elementary matrices.

The tensor product of two matrices can be understood as a block form byconsidering a matrix with as many blocks as the first matrix to betensored has elements, each block being as big as the second matrix tobe tensored. A block of the tensor product is the corresponding elementof the first matrix times the second matrix.

In the above manner we arrive at 2K−2 N-dimensional complexanti-commutator matrices, where N=2^(K−1). The 2K−1th anti-commutatormatrix is obtained by tensoring K−1 third elementary matrices and bymultiplying by the imaginary unit.

Let us next examine an example of the above method. The followinganti-hermitean unitary 2×2 matrices anti-commuting with each other areselected: $\begin{matrix}{{\sigma_{1} = \begin{pmatrix}0 & 1 \\{- 1} & 0\end{pmatrix}};{\sigma_{2} = \begin{pmatrix}0 & i \\i & 0\end{pmatrix}};{\tau = {\begin{pmatrix}{- i} & 0 \\0 & i\end{pmatrix}.}}} & (44)\end{matrix}$

Here the imaginary unit is marked with the letter i. Let us call thepair σ₁, σ₂ an elementary pair and the matrix σ₃=it as a thirdelementary matrix. As a fourth elementary matrix, a 2-dimensional unitmatrix $\begin{matrix}{1_{2} = \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}} & (45)\end{matrix}$

-   -   is used, which is called an elementary unit matrix.

N=2^(K−1)-dimensional complex anti-commutator matrices are formed astensor products of the elementary matrices: $\begin{matrix}{{\gamma_{2} = {\underset{\underset{K - {2\quad{times}}}{︸}}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}} \otimes \sigma_{1}}}{\gamma_{3} = {{{\underset{K - {2\quad{times}}}{\underset{︸}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}}} \otimes \sigma_{2}}\gamma_{4}} = {{{\underset{K - {3\quad{times}}}{\underset{︸}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}}} \otimes \sigma_{1} \otimes \sigma_{3}}\gamma_{5}} = {{{\underset{K - {3\quad{times}}}{\underset{︸}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}}} \otimes \sigma_{2} \otimes \sigma_{3}}\gamma_{6}} = {{{\underset{K - {4\quad{times}}}{\underset{︸}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}}} \otimes \sigma_{1} \otimes \sigma_{3} \otimes \sigma_{3}}\gamma_{7}} = {{{\underset{K - {4\quad{times}}}{\underset{︸}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}}} \otimes \sigma_{2} \otimes \sigma_{3} \otimes \sigma_{3}}\gamma_{2K}} = {\underset{K - 1 - {k\quad{times}}}{\underset{︸}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}}} \otimes \sigma_{1} \otimes \underset{\underset{k - {1\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}}}}}}}{\gamma_{{2K} + 1} = {\underset{\underset{K - 1 - {k\quad{times}}}{︸}}{1_{2} \otimes 1_{2} \otimes 1_{2} \otimes \ldots \otimes 1_{2}} \otimes \sigma_{2} \otimes \underset{\underset{K - 1 - {k\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}}{\gamma_{{2K} - 2} = {\sigma_{1} \otimes \underset{\underset{K - {2\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}}{\gamma_{{2K} - 1} = {\sigma_{2} \otimes \underset{\underset{K - {2\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}}}{\gamma_{1} = \left| {\sigma_{3} \otimes \underset{\underset{K - {1\quad{times}}}{︸}}{\sigma_{3} \otimes \ldots \otimes \sigma_{3}}} \right.}} & (46)\end{matrix}$

The formed matrices are an example of a 2K−1 quantity ofN=2^(K−1)-dimensional antihermitean unitary matrices anticommuting witheach other.

From these matrices, 2K complex anticommutator matrices {γ_(k)+, γ_(k−})_(k=1) ^(K) are formed in the following way: $\begin{matrix}{{\gamma_{k \pm} = \frac{\left( {\gamma_{{2k} - 2} \pm {i\quad\gamma_{{2k} - 1}}} \right)}{2}},{k = 1},\ldots\quad,{K.}} & (47)\end{matrix}$

Matrices γ_(k) defined above are used herein. In addition, the2^(K−1)-dimensional unit matrix is marked with γ₀. The matrices havealso been normalized by dividing by two. The code matrix can now beformed for example as follows: $\begin{matrix}{C = {\sum\limits_{k = 1}^{K}\quad\left( {{s_{k}\quad\gamma_{k -}} + {s_{k}^{*}\quad\gamma_{k +}}} \right)}} & (48)\end{matrix}$

The obtained code is a delay optimal basic block code. All possiblebasic block codes of a given code rate can be created simply byinterchanging the places of rows and/or columns in all γ matricessimultaneously, or by multiplying the γ matrices by any combination ofterms, or changing the numbering of the γ matrices, or by multiplyingall γ matrices from right and/or left by a unitary matrix which has fourelements diverging from zero, the elements being an arbitrarycombination of the numbers ±1, ±i.

For example, the basic rate ¾ code for four transmit antennas as formedin the above manner has the form $\begin{matrix}\left. \left( {s_{1},s_{2},s_{3}} \right)\rightarrow\begin{pmatrix}s_{1} & s_{2} & s_{3} & 0 \\{- s_{2}^{*}} & s_{1}^{*} & 0 & {- s_{3}} \\{- s_{3}^{*}} & 0 & s_{1}^{*} & s_{2} \\0 & s_{3}^{*} & {- s_{2}^{*}} & s_{1}\end{pmatrix} \right. & (49)\end{matrix}$

Here N=4 and K=3. Further, the rate 1/2 code for eight antennas, forexample, is $\begin{matrix}\left. \left( {s_{1},s_{2},s_{3}} \right)\rightarrow\begin{pmatrix}s_{1} & s_{2} & s_{3} & 0 & s_{4} & 0 & 0 & 0 \\{- s_{2}^{*}} & s_{1}^{*} & 0 & {- s_{3}} & 0 & {- s_{4}} & 0 & 0 \\{- s_{3}^{*}} & 0 & s_{1}^{*} & s_{2} & 0 & 0 & {- s_{4}} & 0 \\0 & s_{3}^{*} & {- s_{2}^{*}} & s_{1} & 0 & 0 & 0 & s_{4} \\{- s_{4}^{*}} & 0 & 0 & 0 & s_{1}^{*} & s_{2} & s_{3} & 0 \\0 & s_{4}^{*} & 0 & 0 & {- s_{2}^{*}} & s_{1} & 0 & {- s_{3}} \\0 & 0 & s_{4}^{*} & 0 & {- s_{3}^{*}} & 0 & s_{1} & s_{2} \\0 & 0 & 0 & {- s_{4}^{*}} & 0 & s_{3}^{*} & {- s_{2}^{*}} & s_{1}^{*}\end{pmatrix} \right. & (50)\end{matrix}$

Here the rate ¾ code is in the upper left corner and the correspondinginverted complex conjugate in the lower right corner.

By the above manner, ‘basic codes’ are obtained, in which the elementsonly depend on one signal, or the real part of one signal and theimaginary part of another. The combination of any N′<N code matrixcolumn gives a full diversity non-square code for N′ antennas. Usingthese codes, full diversity codes, which do not have the aboverestriction, can be constructed in the solution of the invention. In asolution according to a preferred embodiment of the invention theelements are allowed to be linear combinations. This way, provided thatfull diversity is provided, block codes that are unitarily converted areobtained, having the form{overscore (C)}=UC(s)V,  (51)

-   -   where C(s) is a basic block code, such as above. It is an N×N′        matrix, where N is the number of time slots and N′ is the number        of antennas. U and V are N×N and N′×N′ unitary matrices. The        phase shifts caused by U and V are irrelevant. U and V can be        assumed to be unitary matrices with determinant 1.

This construction gives a family of block codes with N²+N′²−2 continuousparameters. The square codes obtained this way comprise delay optimalmaximal rate block codes when the number of antennas is proportional toa power of two.

Consider, for example, the rate ¾ code for four antennas which weredescribed above (49). A generic unitary 4×4 matrix with a unitdeterminant can be written, for example, as $\begin{matrix}{V = {\exp\quad\frac{i}{2}\quad\begin{pmatrix}{\Phi_{13} + {\frac{1}{\sqrt{3}}\quad\Phi_{14}} + {\frac{1}{\sqrt{6}}\quad\Phi_{15}}} & w_{1} & w_{2} & w_{3} \\w_{1}^{*} & {{- \Phi_{13}} + {\frac{1}{\sqrt{3}}\quad\Phi_{14}} + {\frac{1}{\sqrt{6}}\quad\Phi_{15}}} & w_{4} & w_{5} \\w_{2}^{*} & w_{4}^{*} & {{\frac{- 2}{\sqrt{3}}\quad\Phi_{14}} + {\frac{1}{\sqrt{6}}\quad\Phi_{15}}} & w_{6} \\w_{3}^{*} & w_{5}^{*} & w_{6}^{*} & {\frac{- 3}{\sqrt{6}}\quad\Phi_{15}}\end{pmatrix}}} & (52)\end{matrix}$

-   -   where the exp operation is a matrix exponential, the six        parameters ω_(k) are complex and the three parameters Φ_(k) are        real. U is of the same form. All in all this makes 30 free real        parameters. All possible generalizations of said ¾ code (49) can        now be constructed by applying the transformation (51) and using        the above described U and V.

It would be desirable that transmission power is distributed equallybetween different antennas. However, when for example the prior art ¾code (the ¾ code mentioned in the introduction of the application, forinstance) is used, some antennas transmit using only half of their powerat certain times. If the code (49) according to a preferred embodimentof the invention is used, the ratio of the peak power to the averagepower can be made lower. In addition, this construction allows thatinstead of transmitting zeros of the code matrix, a signal that isorthogonalized in some other way (for example by a different spreadingcode), a pilot signal for instance, may be transmitted. This way a fullypower-uniformized transmission can be provided.

In a system with several users, especially in a code division andfrequency division system, users may be provided with different versions(for example, a version with a permutated antenna order) of the blockcode, and thus the transmission powers can be uniformized.

Sometimes, in a time division system with several users, for example, itis preferable to balance the transmission of one user directly withoutusing the above mentioned ways. In a solution according to a preferredembodiment of the invention, an unequal distribution of transmissionpower between different antennas may be avoided and the above describedunitary transformation (51) is applied. The power spectrum of differentantennas cannot necessarily be uniformized in respect to each other as afunction of time, but by selecting the unitary transformationpreferable, the average transmission powers of the antennas areuniformized and the ratio of the peak power to the average power and theratio of the minimum power to the average power can be minimized.

Let us examine this embodiment by means of an example. Consider theabove described ¾ code (49) for four antennas. Depending on whichparameter needs to be improved, the matrices U and Vare selected in anappropriate manner. If the minimum-to-average power needs to beoptimized, V is selected as the unit matrix and U as the 4×4 Hadamardmatrix: $\begin{matrix}{U = {\frac{1}{2}\quad\begin{pmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{pmatrix}}} & (53)\end{matrix}$

Now by applying the transformation (51) to the code (49) with the abovementioned matrices U and V, the power-uniformized code is$\begin{matrix}{\overset{\_}{C} = {{{UC}\quad(z)} = {\begin{pmatrix}{s_{1} - s_{2}^{*} - s_{3}^{*}} & {s_{1}^{*} + s_{2}^{*} + s_{3}^{*}} & {s_{1}^{*} - s_{2}^{*} + s_{3}^{*}} & {s_{1} + s_{2} - s_{3}} \\{s_{1} + s_{2}^{*} - s_{3}^{*}} & {{- s_{1}^{*}} + s_{2}^{*} - s_{3}^{*}} & {s_{1}^{*} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}} + s_{2} + s_{3}} \\{s_{1} - s_{2}^{*} + s_{3}^{*}} & {s_{1}^{*} + s_{2}^{*} - s_{3}^{*}} & {{- s_{1}^{*}} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}} - s_{2} - s_{3}} \\{s_{1} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}^{*}} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}^{*}} - s_{2}^{*} + s_{3}^{*}} & {s_{1} - s_{2} + s_{3}}\end{pmatrix}.}}} & (54)\end{matrix}$

On the other hand, if the peak-to-average power needs to be minimized, Uand V can be selected for example as follows: (It is assumed herein thatthe signal constellation is 8-PSK.) $\begin{matrix}{U = {\frac{1}{2}\quad\begin{pmatrix}1 & {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}} & {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}} & {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{3\quad\pi}{8}} \\1 & {- {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}} & {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}} & {- {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{3\quad\pi}{8}}} \\1 & {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}} & {- {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}} & {- {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{3\quad\pi}{8}}} \\1 & {- {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}} & {- {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}} & {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{3\quad\pi}{8}}\end{pmatrix}\quad{and}}} & (55) \\{V = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & {\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}} & 0 & 0 \\0 & 0 & {\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}} & 0 \\0 & 0 & 0 & {\mathbb{e}}^{{\mathbb{i}}\quad\frac{3\quad\pi}{8}}\end{pmatrix}} & (56)\end{matrix}$

Applying now the transformation (51) to the code (49) with the abovematrices U and V, a power-uniformized code is achieved, which has aminimal peak-to-average power $\begin{matrix}{\overset{\_}{C} = {{{UC}\quad(s)\quad V} = {\frac{1}{2}\quad{\begin{pmatrix}{z_{1} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {z_{1}^{*} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {z_{1}^{*} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}} & {z_{1} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} - {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}} \\{z_{1} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {{- z_{1}^{*}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {z_{1}^{*} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}} & {{- z_{1}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}} \\{z_{1} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {z_{1}^{*} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {{- z_{1}^{*}} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}} & {{- z_{1}} - {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}} \\{z_{1} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {{- z_{1}^{*}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} + {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{4}}\quad z_{3}^{*}}} & {{- z_{1}^{*}} - {{\mathbb{e}}^{{- {\mathbb{i}}}\quad\frac{\pi}{8}}\quad z_{2}^{*}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}} & {z_{1} - {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{8}}\quad z_{2}} + {{\mathbb{e}}^{{\mathbb{i}}\quad\frac{\pi}{4}}\quad z_{3}}}\end{pmatrix}.}}}} & (57)\end{matrix}$

Let us next examine a decoding method which can be applied to thereception of the signals that are coded in the above manners. Let usassume that a receiver has M antennas. Let us further assume that N′antennas are used for transmission in a transmitter and that the blockcode uses N time slots. The channel between the nth transmit antenna andthe mth receive antenna is denoted by the term α_(nm). The channels canbe assumed to be static over the frame N. The channel terms arecollected into the N′×M matrix $\begin{matrix}{\alpha = \begin{pmatrix}\alpha_{11} & \alpha_{12} & \ldots & \alpha_{1M} \\\alpha_{21} & \alpha_{22} & \ldots & \alpha_{2M} \\\vdots & \vdots & ⋰ & \vdots \\\alpha_{N1} & \alpha_{N2} & \ldots & \alpha_{NM}\end{pmatrix}} & (58)\end{matrix}$

Correspondingly, the signal received by the antenna m at the time slot tis denoted by r_(tm). The N×M matrix of these signals is obtained fromthe formulaR={overscore (C)}(s)α+n,  (59)

-   -   where n is an N×M matrix of additive complex Gaussian noise. The        block code {overscore (C)} is constructed as above ((1), (2) and        (4)), possibly by restricting the number of antennas. Now denote        {overscore (γ)}_(k±) =U _(γk±) V, k=1 . . . ,K  (60)

Using these markings, the maximum likelihood detection metric for thekth transmitted symbol s_(k) isM _(k) =|Tr({overscore (γ)}_(k+) αR ^(H) +Rα ^(H){overscore (γ)}^(H)_(k−))−s _(k)|²+(Tr(αα^(H))−1)s _(k)|²  (61)

-   -   where Tr refers to a matrix trace, (i.e. the sum of diagonal        elements, and H refers to the complex conjugate transpose).        Thus, the aim is to minimize the metric, (i.e. it is used as a        criterion for deciding which symbol s_(k) comprises).

FIG. 3 is an illustration of an arrangement according to an embodimentof the invention—to be use as exemplar only. The figure shows asituation where channel-coded symbols are transmitted via three antennasat different frequencies, at different time slots or by using adifferent spreading code. Firstly, the figure shows a transmitter 300,which is in connection with a receiver 302. The transmitter comprises amodulator 304 which receives as input a signal 306 to be transmitted,which consists of bits in a solution according to a preferred embodimentof the invention. The bits are modulated to symbols in the modulator.The symbols to be transmitted are grouped into blocks having the lengthof a given K. It is assumed in this example that the length of the blockis three symbols and that the symbols are s₁, s₂ and s₃. The symbols areconveyed to a coder 308. In the coder each block is coded to N×N′channel symbols. The channel symbols 310 are conveyed in this examplevia radio frequency parts 312 to three antennas 314 to 318 from wherethey are to be transmitted.

In the present example, the block comprises the symbols s₁, s₂ and s₃.The coder performs coding, the defining code matrix of which is formedof 2K elements, in which each element is a product of a symbol or symbolcomplex conjugate to be transmitted and a complex N×N anticommutatormatrix, and in which each matrix is used at most once in the formationof the code matrix.

A code matrix can for example be the matrix (54) described above, whichmeans that the coder performs the coding $\begin{matrix}\left. \left( {s_{1},s_{2},s_{3}} \right)\rightarrow{\begin{pmatrix}{s_{1} - s_{2}^{*} - s_{3}^{*}} & {s_{1}^{*} + s_{2}^{*} + s_{3}^{*}} & {s_{1}^{*} - s_{2}^{*} + s_{3}^{*}} & {s_{1} + s_{2} - s_{3}} \\{s_{1} + s_{2}^{*} - s_{3}^{*}} & {{- s_{1}^{*}} + s_{2}^{*} - s_{3}^{*}} & {s_{1}^{*} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}} + s_{2} + s_{3}} \\{s_{1} - s_{2}^{*} + s_{3}^{*}} & {s_{1}^{*} + s_{2}^{*} - s_{3}^{*}} & {{- s_{1}^{*}} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}} - s_{2} - s_{3}} \\{s_{1} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}^{*}} + s_{2}^{*} + s_{3}^{*}} & {{- s_{1}^{*}} - s_{2}^{*} + s_{3}^{*}} & {s_{1} - s_{2} + s_{3}}\end{pmatrix}.} \right. & (62)\end{matrix}$

The coder can preferably be implemented by means of a processor andsuitable software or alternatively by means of separate components.

Let us examine the receiver shown in FIG. 3 once more. By means of thetransmitter of the invention, a signal 320 is transmitted by using threeor more antennas. The signal is received in the receiver 302 by means ofan antenna 322 and it is conveyed to the radio frequency parts 324. Inthe radio frequency parts the signal is converted to an intermediatefrequency or to baseband. The converted signal is conveyed to a channelestimator 326, which forms estimates for the channel through which thesignal has propagated. The estimates can be formed, for example, bymeans of previously known bits the signal contains, such as a pilotsignal or a training sequence code. The signal is conveyed from theradio frequency parts also to a combiner 328, to which also theestimates are delivered from the channel estimator 326. The channelestimator and the radio frequency parts can be implemented by employingthe known methods.

The combiner 328 receives the symbols transmitted at different timeslots, typically stores them temporarily in a buffer memory and formsestimates ŝ_(j), i=1,2,3 for the original block symbols by means of thechannel estimates and the metric (61). A detector 330 performs thesymbol detection according to the formula (61). The signal is conveyedfrom the detector 330 to a channel decoder 332 and further to the otherparts of the receiver. The detector can preferably be implemented bymeans of a processor and suitable software or alternatively by means ofseparate components.

Only one example of a possible receiver is described above. Thecalculation and use of channel estimates, for example, can beimplemented in various other ways, as is obvious to a person skilled inthe art.

For real signal constellations, full rate codes exist for up to 8transmit antennas. $\begin{matrix}{{C_{3/4}\quad\left( {s_{1},s_{2},s_{3}} \right)} = \begin{bmatrix}s_{1} & s_{2} & s_{3} & 0 \\{- s_{2}^{*}} & s_{1}^{*} & 0 & {- s_{3}} \\{- s_{3}^{*}} & 0 & s_{1}^{*} & s_{2} \\0 & s_{3}^{*} & {- s_{2}^{*}} & s_{1}\end{bmatrix}} & (63)\end{matrix}$

The 4-branch MRC linear decoding property of this code is reflected inthe unitarity of the code matrix: $\begin{matrix}{{C_{3/4}^{H}\quad C_{3/4}} = {\left( {{s_{1}\quad s_{1}^{*}} + {s_{2}\quad s_{2}^{*}} + {s_{3}\quad s_{3}^{*}}} \right)\quad I}} & (64)\end{matrix}$

Some features of space-time block codes may be relaxed. One such featureis the requirement of orthogonality of the space-time block code. Hereit is easy to see that rates less than 1 result in power-unbalancedschemes (i.e. the transmit power of a given antenna fluctuates in time).The basic rate ¾ code is given above, the symbols transmitted are s₁,s₂, and s₃. A three antenna version of the code is derived by switchingoff one of the antennas. The inefficiency and power-unbalance is visiblein the zeros of the code. Furthermore, it is clear from the code thatthe inefficiency of the code is due to the existence of redundantdirections in the code matrix, which are not used for signaling. In theabove code, this redundant direction is spanned by the off-diagonal inthe code matrix. This code may be used as the basis of the code providedby another embodiment of the invention a Layered Space-Time Block Codeis identified below. Layered space-time block codes involves patchingpieces of fully orthogonal space-time block codes together.

Again, the symbol-wise maximum likelihood detection metric for thespace-time block code (63) isM _(k) =|Tr(β_(k) +αR ^(H) +Rα ^(H)β_(k−) ^(H))−s_(k)|²+(Tr(α^(H)α)−1)|z _(k)|²  (65)

Here, α is the matrix of channel weights (allowing for a multitude ofreceive antennas), R is the matrix of received symbols, and the decodingmatrices are $\begin{matrix}{{\beta_{1 -} = {{\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\quad\beta_{2 -}} = {{\begin{pmatrix}0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0\end{pmatrix}\quad\beta_{3 -}} = \begin{pmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & {- 1} \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{pmatrix}}}}{\beta_{1 +} = {{\begin{pmatrix}0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 0\end{pmatrix}\quad\beta_{2 +}} = {{\begin{pmatrix}0 & 0 & 0 & 0 \\{- 1} & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & {- 1} & 0\end{pmatrix}\quad\beta_{3 +}} = \begin{pmatrix}0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\{- 1} & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{pmatrix}}}}} & (66)\end{matrix}$

For a space-time block code with rate less than one, the inefficiency ofthe code is exploited to transmit additional data. For example, parts ofanother block code (the higher layer code) may be appended onto theredundant directions of the block code. This is repeated as many time asis required to transmit the full higher layer code.

The upper layers violate the inherent orthogonality of the underlyingblock code, making linear one-step decoding impossible. However, thetwo-layer code can be decoded in two linear steps, first the upperlayer, then, after interference cancellation, the component codes of thelower layer one at a time. Better performance is achieved, if instead oflinear decoding, the upper layer is decoded by a optimal configurationsearch.

Similarly, the possible residual inefficiency of the two-layer code canbe used to transmit a third layer block code. The linear decoding of aI-layer code has to be done in I steps, starting from the uppermostlayer. The interference induced by a higher layer on the lower ones iscancelled after detection of the higher layer. The best performance isachieved when the upper layers are decoded by a optimal configurationsearch.

More mathematically, the basis of the proposed scheme is the following.Consider a space-time block code with N transmit antennas, and forsimplicity, N symbol intervals.

Code matrix orthogonality restricts the number of symbols to some numberK≦N. The proposed scheme applies when K<N. In this case, the antennaresource is used inefficiently, the block code has rate <1, and it ispower-unbalanced.

The received symbol vector is a vector in N-dimensional complex space,call it the reception space. Now one should distinguish between twodifferent kinds of orthogonality. First, there is the orthogonality ofthe code matrix, which translates on reception to linear decodability ofthe block code, and full diversity. Second, there is conventionalorthogonality in the reception space.

The code-matrix-orthogonal K symbols span a K-dimensional subspace ofthe reception space, call it the block-code reception subspace. As K<N,there is room in the reception space for more information. This extrainformation is called the higher layer, and it is encoded into redundantdirections of the code matrix, (i.e. code matrix directions that projectonto directions of the reception space that arereception-space-orthogonal to the block-code reception subspace).

This reception-space-orthogonality does not translate into code-matrixorthogonality, as the latter was already maximized when maximizing K.Thus one-step linear decoding is lost. However, the redundant directionscan be decoded first, and subtracted using interference cancellation.After this, what remains from the received symbols, can be linearlydecoded using code-matrix orthogonality. Optimal decoding of the upperlayer requires searching through the upper layer configurations to findthe optimal one.

Straight forward use of the redundancy does not provide full diversityfor the symbol(s) transmitted on the redundant direction. Rather, thechannels combine uncoherently. To avoid this, one may use a higher layerblock code, so that the outputs of the redundant directions ofsuccessive coding blocks yield the would-be received symbols of a blockcode. This higher-layer block code can then be decoded as a normal blockcode. After interference cancellation, one can decode the lower layerblocks.

Using a block code with rate <1 on the higher layer, leaves someresidual inefficiency, (i.e. the rate of the layered code is still R<1).This inefficiency can again be used to accommodate information symbolsfor a even higher layer code. In this way, the code rate can be madeasymptotically as close to unity as wished.

Alternatively, the additional layering can be terminated on any stage(also before adding any layers to the basic block code), and theremaining inefficiency can be used to transmit known pilot bits. Thesededicated pilots would, apart from improving the channel estimates onreception, make the code completely power-balanced. Also, a rate-1 blockcode (Alamouti 2×2 or no code) can be used on an upper layer to make thelayered code unit rate and power-balanced.

The upper layer codes can be designed so that it is possible toconstruct upper layer pseudo-received signals in which the interferencefrom the lower layers is cancelled completely. These pseudo-receivedsignals are constructed from the received signals and the powers of theestimated channels. The upper-layer code can thus be decoded directly,without considering lower-layer interference. The decoded upper layercan then be cancelled from the received signals, and the lower layer maybe decoded linearly. This decoding scheme is sensitive to changes in thechannels over the whole multi-layer block.

The upper-layer decoding based on pseudo-received symbols issub-optimal, however. An optimal upper-layer decoding would involvesearching the upper layer configuration which after IC and lower-layerdecoding would give the smallest euclidean distance between the receivedsignals and the explained part of the received signals. An optimaldecoding is insensitive to changes in the channels between lowest layersub-blocks.

The optimal decoding can be approached by iterating an IC-procedure. Thetransmitted symbols are divided into two parts, the lowest layer and theupper ones. The detected lowest and upper layer symbols are alternatelysubtracted from the received signals, using the residual signal todetect the upper and lowest layer symbols, respectively.

As the starting point of iterative IC detection, one may use the resultof the multi-step linear decoding. Alternatively, one may start directlyby decoding the lowest layer, with no upper layer IC.

The essential aspects of this embodiment of the invention are:

-   -   1. Using the redundant directions of a space-time block code        matrix for signalling additional data.    -   2. Additional coding of the signals on the redundant direction.    -   3. More specifically, utilizing a higher layer block code for        signaling on the redundant directions of successive lower layer        transmit blocks, to improve the reliability of the higher layer        bits.    -   4. If a redundant block-code is used for the higher layer,        iterating the layering structure any number of times, always        increasing the coding rate, asymptotically making it approach        unity.    -   5. After terminating the layering procedure, possibly using the        remaining redundancy to transmit pilot bits, thus making the        code completely power-balanced.    -   6. Designing the code so that multi-step linear decoding is        possible; with suitable operations, the interference from the        lower layers to an upper ones can be completely cancelled. Thus        one first decodes the highest layer block code, then cancels the        interference from lower layer symbols, and repeats this until        the lowest layer has been decoded. This decoding is sub-optimal,        and sensitive to changes of the channels over the successive        blocks covered by the layering scheme.    -   7. Iterative IC-decoding by alternatingly subtracting the        interference caused by the upper layers on the lowest one and        the lowest layer to the upper ones.        -   The starting point of the first iteration of the lowest            layer might be the upper-layer data produced by the            multi-step linear decoding described above, if the channels            do not change significantly over the whole signaling block.        -   If the channels do change significantly, or if the upper            layer codes have not been designed to enable multi-layer            linear decoding, the iterative IC-decoding should be started            directly by decoding the lowest layer, skipping upper-layer            IC.        -   Iterative IC decoding converges towards an optimal upper            layers decoding by an upper-layers configuration search.

As examples, consider the embodiments built upon the four-antennaspace-time block code C_(3/4).

First we consider a version of the preferred embodiment, where the same4×4 code C_(3/4) is used on all layers. This results in a rate {fraction(15/16)} 2-layer code. A three antenna version of the code is created byturning of one of the antennas. This scheme keeps the diversity gainmaximal by avoiding non-coherent combining loss in the upper layer.

In a two-layer code, the lowest layer of a two-layer code consists offour elementary blocks. Thus the processing delay is 16 symbolintervals. All in all, 15 complex symbols can be encoded into atwo-layer block code with three or four transmit (Tx) antennas.

The redundant direction of a coding block of form (63) is spanned by thematrix $\begin{matrix}{\beta_{0} = \begin{pmatrix}0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 \\0 & {- 1} & 0 & 0 \\{- 1} & 0 & 0 & 0\end{pmatrix}} & (67)\end{matrix}$

This can be seen if R=C({overscore (s)})α+noise to be the receivedsymbol, when the conventional block code equation C_(3/4) above is usedfor transmission, and α is the channel vector. Thereception-space-orthogonality of the block-code reception subspace tothe higher layer reception subspace is reflected in the fact that forthis R,α^(T)β₀ R=0+noise.  (68)

This equation is the basis of the layering embodiment of the invention.

To construct the second layer code, the zeros in the lower layer blocksare filled with rows from the higher layer block code. The higher layermatrix, inserted into the zeros, is constructed from the redundancespanner matrix ,o and a code matrix of the form C_(3/4), namely:$\begin{matrix}{C^{2} = \begin{bmatrix}s_{13} & s_{14} & s_{15} & 0 \\{- s_{14}^{*}} & s_{13}^{*} & 0 & {- s_{15}} \\{- s_{15}^{*}} & 0 & s_{13}^{*} & s_{14} \\0 & s_{15}^{*} & {- s_{14}^{*}} & s_{13}\end{bmatrix}} & (69)\end{matrix}$

To each sub-block of the lower layer, a matrix is added that can beconstructed as the product of the redundance spanner matrix β₀, and amatrix with one of the rows of C² on the diagonal. The resultingtwo-layer block code is shown in FIG. 5 and has a rate R={fraction(15/16)}.

The zeros in the two-layer code in FIG. 4 may now be used to accommodateeither pilot bits, or one row of a third layer block code. Repeatingthis four times, for each row of the Layer 3 block, we get a rate{fraction (63/64)} three-layer block code for four transmit (Tx)antennas. This code provides full diversity, and encodes 63 complexsymbols with a processing delay of 64. An example of a three-layer codeis given in FIG. 5.

The procedure of inserting rows of the third layer code into holes ofthe two-layer codes, differs from inserting rows of the second layer tothe one-layer codes, in that no sign changes indicated by multiplyingwith β₀ are required. This is a generic property—at even levels, some ofthe signs change with β₀, at odd levels, the code matrix rows areinserted as they are.

Repeating the pasting procedure, we get a rate {fraction (255/256)}four-layer, four-antenna code with processing delay 256, a rate{fraction (1023/1024)} five-layer, four-antenna code with processingdelay 1024, and finally a rate 1-2⁻²¹, four-antenna code with I layersand processing delay 2²¹. Corresponding three-antenna codes can becreated by deleting one of the antenna columns.

Finally, if desired, the residual inefficiency of these codes can beused to transmit pilot symbols. This would make the codes completelypower-balanced.

Decoding

The decoding algorithms are presented for one receive (Rx) antenna. Itis easy to generalize to any number of Rx antennas.

An optimal decoding involves an upper layer configuration search. Thus,for example for the 2-layer code of FIG. 4, optimal decoding wouldinvolve going through all 4³=64 combinations of upper layer symbols,canceling the hard bits from the lower layer received signals, decodingthe lower layers using metricM _(k) =|Tr(β_(k) +αR ^(H) +Rα ^(H)β_(k−) ^(H))−s_(k)|²+(Tr(α^(H)α)−1)|s _(k)|²  (70)

-   -   and finding the configuration that minimizes the residual noise        (the absolute difference of the received symbols and the part        explained by the decided bits).

An optimal decoding is insensitive to changes of the channels betweensuccessive lowest layer blocks. The layered scheme thus is as sensitiveto the fading speed as the lowest-layer block code.

This embodiment of the invention provides an iterative IC-procedure thatconverges towards optimal decoding for the 2-layer code in FIG. 4.

The complex channel weights between the n:th transmit and the receiveantenna is denoted α_(n). The channel vector for 4-antenna transmissionis thus α=(α₁; α₂; α₃;α₄). The 4×4 blocks of the two-layer codecorresponding to the lower layer block codes are denoted C({overscore(s)})_(2-layer,m) m=1,2,3,4. Corresponding to these, the receivedsymbols in the sixteen intervals covered by the code may be divided intofour blocks of four:mR=C({overscore (s)})_(2-layer, m)α+noise, m=1,2,3,4.  (71)

To be explicit, the received symbols are $\begin{matrix}{{1R} = {\begin{bmatrix}{{\alpha_{1}\quad s_{1}} + {\alpha_{2}\quad s_{2}} + {\alpha_{3}\quad s_{3}}} \\{{\alpha_{2}\quad s_{1}^{*}} - {\alpha_{1}\quad s_{2}^{*}} - {\alpha_{4}\quad s_{3}} + {\alpha_{3}\quad s_{15}}} \\{{\alpha_{3}\quad s_{1}^{*}} + {\alpha_{4}\quad s_{2}^{*}} - {\alpha_{1}\quad s_{3}^{*}} - {\alpha_{2}\quad s_{14}}} \\{{\alpha_{4}\quad s_{1}} - {\alpha_{3}\quad s_{2}^{*}} + {\alpha_{2}\quad s_{3}^{*}} - {\alpha_{1}\quad s_{13}}}\end{bmatrix} + {noise}}} & (72) \\{{2R} = {\begin{bmatrix}{{\alpha_{1}\quad s_{4}} + {\alpha_{2}\quad s_{5}} + {\alpha_{3}\quad s_{6}} - {\alpha_{4}\quad s_{15}}} \\{{\alpha_{2}\quad s_{4}^{*}} - {\alpha_{1}\quad s_{5}^{*}} - {\alpha_{4}\quad s_{6}}} \\{{\alpha_{3}\quad s_{4}^{*}} + {\alpha_{4}\quad s_{5}} - {\alpha_{1}\quad s_{6}^{*}} - {\alpha_{2}\quad s_{13}^{*}}} \\{{\alpha_{4}\quad s_{4}} - {\alpha_{3}\quad s_{5}^{*}} + {\alpha_{2}\quad s_{6}^{*}} + {\alpha_{1}\quad s_{14}^{*}}}\end{bmatrix} + {noise}}} & (73) \\{{3R} = {\begin{bmatrix}{{\alpha_{1}\quad s_{7}} + {\alpha_{2}\quad s_{8}} + {\alpha_{3}\quad s_{9}} + {\alpha_{4}\quad s_{14}}} \\{{\alpha_{2}\quad s_{7}^{*}} - {\alpha_{1}\quad s_{8}^{*}} - {\alpha_{4}\quad s_{9}} + {\alpha_{3}\quad s_{13}^{*}}} \\{{\alpha_{3}\quad s_{7}^{*}} + {\alpha_{4}\quad s_{8}} - {\alpha_{1}\quad s_{9}^{*}}} \\{{\alpha_{4}\quad s_{7}} - {\alpha_{3}\quad s_{8}^{*}} + {\alpha_{2}\quad s_{9}^{*}} + {\alpha_{1}\quad s_{15}^{*}}}\end{bmatrix} + {noise}}} & (74) \\{{4R} = {\begin{bmatrix}{{\alpha_{1}\quad s_{10}} + {\alpha_{2}\quad s_{11}} + {\alpha_{3}\quad s_{12}} - {\alpha_{4}\quad s_{13}}} \\{{\alpha_{2}\quad s_{10}^{*}} - {\alpha_{1}\quad s_{11}^{*}} - {\alpha_{4}\quad s_{12}} + {\alpha_{3}\quad s_{14}^{*}}} \\{{\alpha_{3}\quad s_{10}^{*}} + {\alpha_{4}\quad s_{11}} - {\alpha_{1}\quad s_{12}^{*}} - {\alpha_{2}\quad s_{15}^{*}}} \\{{\alpha_{4}\quad s_{10}} - {\alpha_{3}\quad s_{11}^{*}} + {\alpha_{2}\quad s_{12}^{*}}}\end{bmatrix} + {noise}}} & (75)\end{matrix}$

With the interference from the upper layer symbols s₁₃; s₁₄; s₁₅(partly) cancelled, each of these subblocks can be decoded as a 4×4block code of the form C_(3/4), using metric (65) above.

Conversely, with the interference from the lowest layer cancelled, onecan see from equations (72-75) that each received signal carriesinformation of only one of the upper layer symbols, multiplied by achannel. It is clear how these can be maximal ratio combined bymultiplying with suitable channel estimates. Due to the space-time blockcode structure of the upper layer, MRC gives four-diversity-branchcombined estimates for the symbols.

For slow-fading channels, the linear I-step decoding described in thenext Section, can be used as a starting point for iterative ICdetection, making convergence faster. For fast fading channels, oneshould start the iterations by first detecting the lowest layer with noIC.

Linear I-Step Decoding

−α₁s₁₃

An embodiment of the invention provides for a detection and interferencecancellation procedure, which decodes in I-linear steps the preferredembodiment I-layered code built on space-time block code C_(3/4). Thechannels are supposed to be almost static over the I-layer processingperiod of length 2²¹.

In addition to the estimated channel, we shall also need the vector ofsquared channels α²=(α₁ ²,α₂ ²,α₃ ²,α₄ ²), and for the I-layer code, thevector of fourth powers of channels, α⁴=(α₁ ⁴,α₂ ⁴,α₃ ⁴, α₄ ⁴), up tothe vector of 2I-2:th powers of channels α^((2I−2)=α) ^(2I−2)=(α₁^(2I−2),α₂ ^(2I−2),α₃ ^(2I−2),α₄ ^(2I−2)).

First, the two-layer code constructed in FIG. 4 is decoded.

From each set _(m)R of four received symbols in above equations (72-75),we can construct a second layer pseudo-received symbol,R _(m) ⁽²⁾=−α^(T)β_(0 m) R.  (76)Here we see how the data encoded onto the redundant direction of thecode-matrix C_(3/4) is extracted from the received signals.

The pseudo-received symbols constructed in (70) carry the data encodedonto the redundant directions of the four first layer blocks. The vectorof these pseudo-received symbols is $\begin{matrix}{R^{2} = {\left\lbrack \quad\begin{matrix}{{\alpha_{1}^{2}s_{13}} + {\alpha_{2}^{2}s_{14}} + {\alpha_{3}^{2}s_{15}}} \\{{\alpha_{2}^{2}s_{13}^{*}} - {\alpha_{1}^{2}s_{14}^{*}} - {\alpha_{4}^{2}s_{15}}} \\{{\alpha_{3}^{2}s_{13}^{*}} + {\alpha_{4}^{2}s_{14}} - {\alpha_{1}^{2}s_{15}^{*}}} \\{{\alpha_{4}^{2}s_{13}} - {\alpha_{3}^{2}s_{14}^{*}} + {\alpha_{2}^{2}s_{15}^{*}}}\end{matrix}\quad \right\rbrack + {noise}}} & (77)\end{matrix}$Now one can employ a variant of the metric given above in (65) to decodethe symbols s₁₃, s₁₄ and s₁₅. The ensuing decoding metric isM _(k) ² =|R ^((2)H)β_(k+)α⁽²⁾+α^((2)H)β_(k−) ^(H) R ⁽²⁾ −s_(k)|²+(α^((2)H)α⁽²⁾−1)|s _(k) ²  (78)which gives an estimate for S_(12+k), when the detection matrices β_(k±)are used.

These estimates should now be subtracted from the received symbols. Tominimize interference, on may compute (e.g. the maximum a posteriori(MAP) estimates of the transmitted higher layer symbols, {ŝ₁₃,ŝ₁₄,ŝ₁₅},and of the higher layer code matrix $\begin{matrix}{{\hat{C}}^{2} = {\left\lbrack \quad\begin{matrix}{\hat{s}}_{13} & {\hat{s}}_{14} & {\hat{s}}_{15} & 0 \\{- {\hat{s}}_{14}^{*}} & {\hat{s}}_{13}^{*} & 0 & {- {\hat{s}}_{15}} \\{- {\hat{s}}_{15}^{*}} & 0 & {\hat{s}}_{13}^{*} & {\hat{s}}_{14} \\0 & {\hat{s}}_{15}^{*} & {- {\hat{s}}_{14}^{*}} & {\hat{s}}_{13}\end{matrix}\quad \right\rbrack.}} & (79)\end{matrix}$

Denoting the matrices with rows of this matrix on the diagonal as Ĉ_(m)^((2,diag)), m=1,2,3,4, we can cancel the interference from the blocksof four successive received symbols (71) above:m{tilde over (R)}=mR−β ₀ Ĉ _(m) ^((2,diag))α.  (80)These IC-cancelled blocks {tilde over (R)}_(m) can now be decoded asconventional space-time blocks codes using metric (65).

The interference is only one-way; the upper layer bits disturb the lowerlayer bits, whereas the upper layer does not see the lower layer.

For a code with three or more layers, this procedure is iterated thecorresponding number of times, always starting from the highest layer.

Thus, for the three layer code shown in FIG. 5, one first divides the 64received symbols into 16 blocks of 4, _(m)R, m=1, . . . ,16. For each ofthese blocks, a second layer pseudo-received symbol R_(m) ⁽²⁾ isconstructed as described above. These are collected into four blocks offour, _(n)R⁽²⁾, N=1, . . . ,4. Each of these blocks are of form (77),plus third layer interference.

From each set _(n)R⁽²⁾ of the four second-layer pseudo-received symbols,we can construct a third-layer pseudo-received symbol,R _(n) ⁽³⁾=−α^((2)Tβ) ^(0 n) ^(R) ⁽²⁾ .  (81)These carry the data encoded onto the redundant directions of the foursecond layer blocks, each consisting of four first layer blocks. To beexplicit, the vector of these pseudo-received symbols is $\begin{matrix}{R^{3} = {\left\lbrack \quad\begin{matrix}{{\alpha_{1}^{4}s_{61}} + {\alpha_{2}^{4}s_{62}} + {\alpha_{3}^{4}s_{63}}} \\{{\alpha_{2}^{4}s_{61}^{*}} - {\alpha_{1}^{4}s_{62}^{*}} - {\alpha_{4}^{4}s_{63}}} \\{{\alpha_{3}^{4}s_{61}^{*}} + {\alpha_{4}^{4}s_{62}} - {\alpha_{1}^{4}s_{63}^{*}}} \\{{\alpha_{4}^{4}s_{61}} - {\alpha_{3}^{4}s_{62}^{*}} + {\alpha_{2}^{4}s_{63}^{*}}}\end{matrix}\quad \right\rbrack + {noise}}} & (82)\end{matrix}$This again may be decoded with the metric provided above. Thus,M _(k) ³ =|R ^((3)H)β_(k+)α⁽⁴⁾+α^((4)H)β_(k−) ^(H) R ⁽³⁾ −s_(k)|²+(α^((4)H)α⁽⁴⁾−1)|s _(k)|²  (83)which gives an estimate for s_(60+k), when the detection matricesβ_(k±)are used.

These estimates should be subtracted from the second layerpseudo-received symbols collected into the blocks _(n)R⁽²⁾. Afterthird-layer interference cancellation, each of these blocks may bedecoded with the metric (78), and again, after second-layer IC, thesixteen first-layer blocks may be decoded using metric (65).

The decoding of the I:th layer proceeds in a similar way, only that inconstructing the I:th layer pseudo-received symbols, as in (76, 81), thechannel vector α^((2I−2)) is used, and in the I:th layer metric,corresponding to (78, 83), the channel vector α^((2I−2)) is used.

By employing a different upper layer code, the rate of a layered codecan be increased even more than by using the same code on all layers.The price to pay for this is performance losses due to non-coherentcombining of upper layer channels.

Specifically, if one uses a full rate upper layer code, i.e. theAlamouti 2×2 block code, or just transmits one symbol on the upperlayer, one gets completely power-balanced block codes with unit rate.

The simplest rate 1 code is the one where no code is used at the upperlayer. That is, four complex symbols are encoded as $\begin{matrix}{{C_{{layered}\quad 4 \times 4}\left( {s_{1},s_{2},s_{3},s_{4}} \right)} = \left\lbrack \quad\begin{matrix}s_{1} & s_{2} & s_{3} & s_{4} \\{- s_{2}^{*}} & s_{1}^{*} & s_{4} & {- s_{3}} \\{- s_{3}^{*}} & {- s_{4}} & s_{1}^{*} & s_{2} \\{- s_{4}} & s_{3}^{*} & {- s_{2}^{*}} & s_{1}\end{matrix}\quad \right\rbrack} & (84)\end{matrix}$A two-step linear decoding of this code starts with creating a secondlayer pseudo-received symbol, as in Equation (76), then making a symboldecision for s₄ based upon this pseudo-received symbol, subtracting theinterference from the lower layer, (i.e. s₁; s₂; s₃), and decoding themas a normal block code.

An optimal decoding procedure involves decoding the lower layerseparately using metric (65) for each of the 4 possible upper layer (s₄)configurations and choosing the one which minimizes residual noise, orjust searching the 4⁴=256 configurations of the full code for theoptimal one.

The absolute square of the above matrix isC _(layered) 4×4^(H) C _(layered) 4×4=(s ₁ s ₁ *+s ₂ s ₂ *+s ₃ s ₃ *+s ₄s ₄*), I+N  (85)

-   -   where the non-orthogonality of the matrix N is $\begin{matrix}        {N = {{2\left\lbrack \quad\begin{matrix}        0 & {{Re}\left\lbrack {s_{3}s_{4}} \right\rbrack} & {- {{Re}\left\lbrack {s_{2}s_{4}} \right\rbrack}} & {{Re}\left\lbrack {s_{1}^{*}s_{4}} \right\rbrack} \\        {- {{Re}\left\lbrack {s_{3}s_{4}} \right\rbrack}} & 0 & {{Re}\left\lbrack {s_{1}s_{4}} \right\rbrack} & {{Re}\left\lbrack {s_{2}^{*}s_{4}} \right\rbrack} \\        {{Re}\left\lbrack {s_{2}s_{4}} \right\rbrack} & {- {{Re}\left\lbrack {s_{1}s_{4}} \right\rbrack}} & 0 & {{Re}\left\lbrack {s_{3}^{*}s_{4}} \right\rbrack} \\        {{Re}\left\lbrack {s_{1}^{*}s_{4}} \right\rbrack} & {{Re}\left\lbrack {s_{2}s_{4}^{*}} \right\rbrack} & {{Re}\left\lbrack {s_{3}s_{4}^{*}} \right\rbrack} & 0        \end{matrix}\quad \right\rbrack}.}} & (86)        \end{matrix}$

As a measure for non-orthogonality μ_(no), the magnitude of theoff-diagonal elements are compared to the magnitude of the diagonalelements in the hermitean square (85) of the code matrix. More exactly,the expectation value of the ratio of the sum of the absolute squares ofthe off-diagonal elements to the sum of the absolute squares of thediagonal elements. This expectation value is taken over the set ofcodeword pair differences, i.e. over the set of all possible sets offour symbol difference, where the symbols are in the alphabet inquestion. In the preferred embodiment of the invention, this is QPSK.$\begin{matrix}{\mu_{no} = {\frac{1}{4}E\left\langle \left( \frac{{N}_{2}}{\sum\limits_{i = 1}^{4}\quad{s_{i}}^{2}} \right)^{2} \right\rangle}} & (87)\end{matrix}$

Thus, for the layered method and apparatus, the non-orthogonalitymeasure is ⅜=0.375, (if QPSK modulation is used).

A unit-rate 2-layer code with better performance than the one above (butwith double delay), can be built if the Alamouti code (3) is used on theupper layer. Thus, eight complex symbols would be encoded e.g. as alayered 8×4 block code with rate 1 given by $\begin{matrix}{{C_{{layered}\quad 8 \times 4}\left( {s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7},s_{8}} \right)} = \left\lbrack \quad\begin{matrix}s_{1} & s_{2} & s_{3} & s_{4} \\{- s_{2}^{*}} & s_{1}^{*} & s_{8} & {- s_{3}} \\{- s_{3}^{*}} & {- s_{7}} & s_{1}^{*} & s_{2} \\{- s_{7}} & s_{3}^{*} & {- s_{2}^{*}} & s_{1} \\s_{4} & s_{5} & s_{6} & s_{7}^{*} \\{- s_{5}} & s_{4}^{*} & s_{7}^{*} & {- s_{6}} \\{- s_{6}^{*}} & s_{8}^{*} & s_{4}^{*} & s_{5} \\s_{8}^{*} & s_{6}^{*} & {- s_{5}^{*}} & s_{4}\end{matrix}\quad \right\rbrack} & (88)\end{matrix}$

In a 2-step linear decoding procedure, the eight received signals aresplit into two groups of four, and from each group a pseudo-receivedsymbol is constructed as in Equation (76). These are decoded as Alamoutiblock-coded symbols in the “squared” channel α²=(α₁ ²,α₂ ²,α₃ ²,α₄ ²)analogously to the decoding procedure for the rate—{fraction (15/16)}code. Interference is cancelled from the two lower layer codes, andthese are decoded as a normal 4×4 block code, using the metric (65).

An optimal decoding involves searching through the 16 upper layerconfigurations, canceling the interference of these, decoding the lowerlayer using metric (65), and choosing the configuration which minimizesthe residual noise.

The non-orthogonality determined by the procedure set out above measures{fraction (3/16)} or 0.1875. The layered block code method is aniterative decoding that is based on interference cancellation betweenthe fully orthogonal pieces of the layered code. It rapidly approachesthe optimal performance provided by a maximum likelihood decoder. Theco-pending application also provides a non-orthogonal four-antennaspace-time block code based on a matrix of size 16×4, (i.e. it had aprocessing delay of 16 symbol periods or time epochs), with a rate equalto {fraction (15/16)}. The non-orthogonality of the code measures{fraction (2/25)} or 0.08.

With constant-amplitude modulation, it is possible to designpower-balanced codes so that the transmitted signal from each antennahas a constant power envelope. The matrix elements in the code matricesof such codes depend on only one modulation symbol. FIGS. 6A and 6B showthe two forms of matrices from which a power-balanced full rate code maybe constructed.

The symbol types, the first occurrence of which have reference numbers610, 620, 630, 640, means that the corresponding matrix element dependson one symbol or its complex conjugate only. Moreover, for constantenvelope symbols, all matrix elements have the same norm. This isnatural from the full raw diversity point of view. Thus, any of thehollow circles 610 is some phase factor times s₁ or s₁*; any of thestars 620 is some phase factor times s₂ or s₂*; any of the solid circles630 is some phase factor times s₃ or s₃*; and any of the diamonds 640 issome phase factor times s₄ or s₄*. It should be noted that in such anencoding, the germinal matrices, (i.e. the parts of the code matrixdepending only on one of the symbols, are all unitary matrices bydefault).

It is easy to convince oneself that any power-balanced full rate, fullraw diversity four-antenna code can be mapped to one of the two formsshown in FIGS. 6A and 6B by discrete matrix operations likeinterchanging columns and/or rows. These operations do not affectperformance properties. Also, generic unitary transformations may beapplied from left and/or right, changing the power spectrum, but leavingthe performance unchanged.

Several versions of the 4×4 code with μ_(no)=0.25 may be found. Twocopies of the Alamounti block code or Radon-Hurwitz sub-matrix may beused to generate Space-Time Codes for more than 2 antennas, again thismatrix form is given by: $\begin{matrix}{Z_{12} = \begin{bmatrix}z_{1} & z_{2} \\{- z_{2}^{*}} & z_{1}^{*}\end{bmatrix}} & (89)\end{matrix}$

For minimal non-orthogonality, the four symbols may be arranged asfollows replacing the complex (z) representation in the generalsub-matrix form with (s) to represent the symbol to be transmitted.$\begin{matrix}{{C_{ABBA}\left( {s_{1},s_{2},s_{3},s_{4}} \right)} = {\left\lbrack \quad\begin{matrix}s_{1} & s_{2} & s_{3} & s_{4} \\{- s_{2}^{*}} & s_{1}^{*} & {- s_{4}^{*}} & s_{3}^{*} \\s_{3} & s_{4} & s_{1} & s_{2} \\{- s_{4}^{*}} & s_{3}^{*} & {- s_{2}^{*}} & s_{1}^{*}\end{matrix}\quad \right\rbrack.}} & (89)\end{matrix}$

C_(ABBA) has two of the 2×2 Alamouti block code with symbols s₁ and s₂on the block diagonal, and has two copies of the Alamounti code withsymbols s₃, s₄ on the block anti-diagonal in the form $\begin{matrix}{\begin{bmatrix}A & B \\B & A\end{bmatrix}.} & (91)\end{matrix}$

This embodiment may be referred to as the “ABBA” scheme. By taking(s₃,s₄)=(s₁,s₂) during the first two time epochs T₁ and T₂, and (s₁,s₂)=s₃,s₄) during the last two T₃ and T₄ a transmission to an user with3GPP release 99 equipment may be realized, provided that the commonchannel pilots are organized in a suitable way. Thus, providing forbackward compatibility to 3GPP release 99 open-loop diversity.

The non-orthonormality of this embodiment isC _(ABBA) ^(H) C _(ABBA)=(s ₁ s ₁ *+s ₂ s ₂ *+s ₃ s ₃ *+s ₄ s₄*)I+N  (92)

-   -   where the non-orthogonality matrix N is $\begin{matrix}        {N = {2{{{jIm}\left\lbrack {{s_{1}s_{3}^{*}} + {s_{2}s_{4}^{*}}} \right\rbrack}\left\lbrack \quad\begin{matrix}        0 & 0 & 1 & 0 \\        0 & 0 & 0 & 1 \\        1 & 0 & 0 & 0 \\        0 & 1 & 0 & 0        \end{matrix}\quad \right\rbrack}}} & (93)        \end{matrix}$

This gives ¼ for the measure of non-orthogonality. The symbols so and s₂are orthogonally encoded with respect to each other, as are s₃ and s₄.All non-orthogonality is between these two sets. A three Tx-antennaversion may be obtained by turning off one of the antennas.

For detection, the interference caused by symbols on each other, relatedto the non-orthonormality, has to be cancelled. The simple structure ofthe non-orthonormality matrix results in an easily implementable LeastMinimal Mean Squares Estimate (LMMSE) decoding. Here, decoding in aone-path channel is considered. Multi-paths may be treated with knownequalization methods.

As stated above, the channel is given by h_(k)=α_(k)e^(jθ) ^(k) ∀k=1 toK. With K=4 for the four Tx antennas, the channel coefficients areα=[α₁, α₂, α₃, α₄], the received symbol vector isR=C _(ABBA)α+noise.  (94)

For decoding, the following channel matrix may be used: $\begin{matrix}{H = \left\lbrack \quad\begin{matrix}\alpha_{1} & \alpha_{2} & \alpha_{3} & \alpha_{4} \\\alpha_{2}^{*} & {- \alpha_{1}^{*}} & \alpha_{4}^{*} & {- \alpha_{3}^{*}} \\\alpha_{3} & \alpha_{4} & \alpha_{1} & \alpha_{2} \\\alpha_{4}^{*} & {- \alpha_{3}^{*}} & \alpha_{2}^{*} & {- \alpha_{1}^{*}}\end{matrix}\quad \right\rbrack} & (95)\end{matrix}$

With this, we may construct linear estimates of the symbols:$\begin{matrix}{{H\left\lbrack \quad\begin{matrix}R_{1} \\R_{2}^{*} \\R_{3} \\R_{4}^{*}\end{matrix}\quad \right\rbrack} = {{\left( {{\left( {{\alpha_{1}\alpha_{1}^{*}} + {\alpha_{2}\alpha_{2}^{*}} + {\alpha_{3}\alpha_{3}^{*}} + {\alpha_{4}\alpha_{4}^{*}}} \right)l} + \overset{\sim}{N}} \right)\left\lbrack \quad\begin{matrix}s_{1} \\s_{2} \\s_{3} \\s_{4}\end{matrix}\quad \right\rbrack} + {{noise}.}}} & (96)\end{matrix}$

One may use the Real or Imaginary parts of the code. Therefore, thecorrelation matrix was denoted $\begin{matrix}{{{N = {2{{{jIm}\left\lbrack {{s_{1}s_{3}^{*}} + {s_{2}s_{4}^{*}}} \right\rbrack}\left\lbrack \quad\begin{matrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{matrix}\quad \right\rbrack}}},{or}}{N = {2\quad{{{{Re}\left\lbrack {{s_{1}s_{3}^{*}} + {s_{2}s_{4}^{*}}} \right\rbrack}\left\lbrack \quad\begin{matrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{matrix}\quad \right\rbrack}.}}}} & (97)\end{matrix}$

This matrix may be easily inverted. Leading to a Least Minimal MeanSquares Estimate (LMMSE) decoder. The decorrelating inverse of thecorrelation matrix may be taken with $\begin{matrix}{{D = {I - {\frac{a}{1 - a^{2}}\left\lbrack \quad\begin{matrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{matrix}\quad \right\rbrack}}},{where}} & (98) \\{\quad{{{a = \frac{2{jIm}\left\lfloor {{\alpha_{1}\alpha_{3}^{*}} + {\alpha_{2}\alpha_{4}^{*}}} \right\rfloor}{{\alpha_{1}\alpha_{1}^{*}} + {\alpha_{2}\alpha_{2}^{*}} + {\alpha_{3}\alpha_{3}^{*}} + {\alpha_{4}\alpha_{4}^{*}} + b}},{or}}\quad{a = \frac{2{Re}\left\lfloor {{\alpha_{1}\alpha_{3}^{*}} + {\alpha_{2}\alpha_{4}^{*}}} \right\rfloor}{{\alpha_{1}\alpha_{1}^{*}} + {\alpha_{2}\alpha_{2}^{*}} + {\alpha_{3}\alpha_{3}^{*}} + {\alpha_{4}\alpha_{4}^{*}} + b}}}} & (99)\end{matrix}$

With different values of b, different estimates for the symbols may beconstructed. Decorrelation estimates may be found by putting b=0. Thus,$\begin{matrix}{{{D\left( {b = 0} \right)}{H\left\lbrack \quad\begin{matrix}R_{1} \\R_{2}^{*} \\R_{3} \\R_{4}^{*}\end{matrix}\quad \right\rbrack}} = {{\left( {{\left( {{\alpha_{1}\alpha_{1}^{*}} + {\alpha_{2}\alpha_{2}^{*}} + {\alpha_{3}\alpha_{3}^{*}} + {\alpha_{4}\alpha_{4}^{*}}} \right)l} + \overset{\sim}{N}} \right)\left\lbrack \quad\begin{matrix}s_{1} \\s_{2} \\s_{3} \\s_{4}\end{matrix}\quad \right\rbrack} + {{noise}.}}} & (100)\end{matrix}$

With b, the sum of the Signal-to-Noise Ratios (SNR)s over the fourchannels, we get a LMMSE estimate.

As is the case in multi-user interference cancellation and as observedabove regarding layered Space-Time Codes, iterative methods may be usedto improve the estimates of correlating symbols. Thus, one may subtractfrom the received symbols the estimates of symbols s₃ and s₄, thendetect the (orthogonally encoded) symbols s₁ and s₂, and vice versa.Moreover, in concordance with IC results for low interference, most ofthe benefit may be reaped with one re-iteration only.

Other embodiments of the ABBA code can now be constructed simply byreplacing any number of the 2×2 blocks in the ABBA code by anotherequivalent 2×2 block, depending on the same two symbols.

Another way to build embodiments of the invention is to restrict thenon-orthogonality to be block diagonal instead of the block off-diagonalnon-orthogonality (97) of the preferred embodiment. Thus, one can takefor example $\begin{matrix}{{A\left( {{s1},{s2}} \right)} = \begin{bmatrix}s_{1} & s_{2} \\{\pm s_{2}} & {\pm s_{1}}\end{bmatrix}} & (101)\end{matrix}$

-   -   and build a code with the upper two blocks in ABBA being of the        above form (101), and the lower two blocks being complex        conjugates of the above form (101), e.g. $\begin{matrix}        {C_{{ABBA}_{({s_{1},s_{2},s_{3},s_{4}})}} = \left\lbrack \quad\begin{matrix}        s_{1} & s_{2} & s_{3} & s_{4} \\        {- s_{2}} & s_{1} & s_{4} & {- s_{3}} \\        {- s_{3}^{*}} & {- s_{4}^{*}} & s_{1}^{*} & s_{2}^{*} \\        {- s_{4}^{*}} & s_{3}^{*} & {- s_{2}^{*}} & s_{1}^{*}        \end{matrix}\quad \right\rbrack} & (102)        \end{matrix}$

The generalization of the ABBA code to five or to eight antennas is atwice iterated ABBA: $\begin{matrix}{C = \left\lbrack \quad\begin{matrix}A & B & C & D \\B & A & D & C \\C & D & A & B \\D & C & B & A\end{matrix}\quad \right\rbrack} & (103)\end{matrix}$

-   -   where A; B; C; D are of the form (89), and depend on the symbols        (s₁;s₂), (s₃;s₄), (s₅;s₆), (s₇;s₈) respectively. The        non-orthogonality of this scheme measures to ⅜=0.375. Also, the        non-orthogonality matrix is scarce, so it can easily be inverted        to produce a reliable LMMSE decoding scheme.

This code should be contrasted with the generalization of delay optimallayered codes to 8 dimensions which can be constructed from theorthogonal full diversity 8 antenna block code $\begin{matrix}{\quad{{C_{{layered}\quad 4 \times 4}\left( {s_{1},s_{2},s_{3},s_{4}} \right)} = \begin{bmatrix}s_{1} & s_{2} & s_{3} & s_{4} \\{- s_{2}^{*}} & s_{1}^{*} & s_{4} & {- s_{3}} \\{- s_{3}^{*}} & {- s_{4}} & s_{1}^{*} & s_{2} \\{- s_{4}} & s_{3}^{*} & {- s_{2}^{*}} & s_{1}\end{bmatrix}}} & (104) \\{{C_{1/2}\left( {s_{1},s_{2},{s_{3,}s_{4}}} \right)} = \begin{bmatrix}s_{1} & s_{2} & s_{3} & 0 & s_{4} & 0 & 0 & 0 \\{- s_{2}^{*}} & {- s_{1}^{*}} & 0 & {- s_{3}} & 0 & {- s_{4}} & 0 & 0 \\{- s_{3}^{*}} & 0 & s_{1}^{*} & s_{2} & 0 & 0 & {- s_{4}} & 0 \\0 & s_{3}^{*} & {- s_{2}^{*}} & s_{1} & 0 & 0 & 0 & s_{4} \\{- s_{4}^{*}} & 0 & 0 & 0 & s_{1}^{*} & s_{2} & s_{3} & 0 \\0 & s_{4}^{*} & 0 & 0 & {- s_{2}^{*}} & s_{1} & 0 & {- s_{3}} \\0 & 0 & s_{4}^{*} & 0 & {- s_{3}^{*}} & 0 & s_{1} & s_{2} \\0 & 0 & 0 & {- s_{4}^{*}} & 0 & s_{3}^{*} & {- s_{2}^{*}} & s_{1}^{*}\end{bmatrix}} & (105)\end{matrix}$by overlaying it with a version of itself. The result is apower-balanced non-orthogonal code with rate 1 $\begin{matrix}{{{C\left( \quad \right.}\quad s_{1}},\quad s_{2},\quad s_{3},\quad s_{4},\quad s_{5},\quad s_{6},\quad{\left. \quad{s_{7},\quad s_{8}} \right) = {\quad\begin{bmatrix}s_{1} & s_{2} & s_{3} & s_{5} & s_{4} & s_{6} & s_{7} & s_{8} \\{- s_{2}^{*}} & {- s_{1}^{*}} & {- s_{5}} & {- s_{3}} & {- s_{6}} & {- s_{4}} & s_{8}^{*} & {- s_{7}^{*}} \\{- s_{3}^{*}} & {- s_{5}} & s_{1}^{*} & s_{2} & s_{7} & s_{8}^{*} & {- s_{4}} & {- s_{6}^{*}} \\s_{5} & s_{3}^{*} & {- s_{2}^{*}} & s_{1} & s_{8} & {- s_{7}^{*}} & s_{6}^{*} & s_{4} \\{- s_{4}^{*}} & s_{6} & s_{7} & s_{8}^{*} & s_{1}^{*} & s_{2} & s_{3} & {- s_{5}^{*}} \\{- s_{6}^{*}} & s_{4}^{*} & s_{8} & {- s_{7}^{*}} & {- s_{2}^{*}} & s_{1} & s_{5}^{*} & {- s_{3}} \\s_{7} & s_{8} & s_{4}^{*} & {- s_{6}^{*}} & {- s_{3}^{*}} & s_{5}^{*} & s_{1} & s_{2} \\s_{8}^{*} & {- s_{7}^{*}} & s_{6}^{*} & {- s_{4}^{*}} & {- s_{5}^{*}} & s_{3}^{*} & {- s_{2}^{*}} & s_{1}^{*}\end{bmatrix}}}} & (106)\end{matrix}$This code has non-orthogonality ½. Thus, the twice iterated ABBA code(103) may be superior to this scheme, and similarly superior to anyscheme in which the code matrix is put together from two maximallyorthogonal pieces.

Therefore, indicating that for any number of antennas, delay optimalminimally non-orthogonal space-time block codes may be constructed byrepeatedly iterating the ABBA method.

An embodiment of the invention provides a class of sub-optimalspace-time block codes. Referring back to the Radon-Hurwitz matrix$Z_{12} = {\begin{bmatrix}z_{1} & z_{2} \\{- z_{2}^{*}} & z_{1}^{*}\end{bmatrix}.}$It is a structured matrix because it may be filled with any complex pairof symbols. Also, this structured matrix may be used as a sub-block toconstruct structured matrix classes or families. Because there is directrelation between the Z matrix or structure and the $H = {\begin{bmatrix}h_{1} & h_{2} \\h_{2}^{*} & {- h_{1}^{*}}\end{bmatrix}.}$matrix, the attention will be focused on the latter one. Subsequently, afew proprieties of H matrix are given below. The following matrixmultiplication plays an important role in the class of space-time codesprovide by the present invention: $\begin{matrix}{{{{H_{12}^{H}H_{34}} = \begin{bmatrix}x & y \\{- y^{*}} & x^{*}\end{bmatrix}},{where}}{{x = {{h_{1}^{*}h_{3}} + {h_{2}h_{4}^{*}}}},{{{and}\quad y} = {{h_{1}^{*}h_{4}} - {h_{2}{h_{3}^{*}.}}}}}} & (107)\end{matrix}$

Note that $\begin{matrix}{{{H_{12}^{H}H_{34}} + {H_{34}^{H}H_{12}}} = {\begin{bmatrix}{x + x^{*}} & 0 \\0 & {x + x^{*}}\end{bmatrix}\quad{and}}} & (108) \\{{{H_{12}^{H}H_{34}} - \left( {H_{34}^{H}H_{12}} \right)^{*}} = {\begin{bmatrix}0 & {y + y^{*}} \\{- \left( {y + y^{*}} \right)} & 0\end{bmatrix}.}} & (109)\end{matrix}$

Denoting by L the number of transmitting antennas and by “{circle around(×)}” the tensor product, the equivalent general form H matrix of theproposed space-time block code (STC) is given by $\begin{matrix}{H = {\sum\limits_{k = 1}^{L/2}{\left( {{C_{{{2k} - 1},{2k}} \otimes K_{{{2k} - 1},{2k}}} + {D_{{{2k} - 1},{2k}} \otimes H_{{{2k} - 1},{2k}}^{*}}} \right).}}} & (110)\end{matrix}$

C_(2k−1,2k) and D_(2k−1,2k) are p×p matrices that:

-   -   have at most one nonzero entry for each row and column;    -   any two matrices cannot have a nonzero entry in the same        position, (i.e. the matrices have disjoint nonzero entries);    -   a nonzero entry is modulus one (unit power);    -   a row in the H matrix is filled either by H_(2k−1,2k) or by        H*_(2k−1,2k) matrices.

This ensures that a given symbol is assigned only once to a givenantenna and at a given time instant is sent by one antenna. Also, thereis a one-to-one correspondence between H and S matrices.

For the purpose of presentation we denote by H_(k), C_(k) and D_(k) thematrices:

-   -   H_(2k−1,2k),    -   C_(2k−1,2k), and    -   D_(2k−1,2k), respectively.

Using (13), the general form of (5) is:r=Hs+n  (111)

In the detection process, matched filtering is performed, (i.e.left-side multiplication of (14) by the H^(H) matrix). Using the factthat:C ^(H) _(k) D _(j) =D ^(H) _(k) C _(j)=0  (112)

-   -   this multiplication yields: $\begin{matrix}        \begin{matrix}        {H^{(1)} = {{H^{H}H} = {{\sum\limits_{i = 1}^{L/2}{\left( {C_{i}^{H}C_{i}} \right) \otimes \left( {H_{i}^{H}H_{i}} \right)}} +}}} \\        {{\sum\limits_{\underset{i \neq j}{i,{j = 1}}}^{L/2}{\left( {C_{i}^{H}C_{j}} \right) \otimes \left( {H_{i}^{H}H_{j}} \right)}} + {\sum\limits_{i = 1}^{L/2}{\left( {D_{i}^{H}D_{i}} \right) \otimes \left( {H_{i}^{H}H_{i}} \right)}} +} \\        {\sum\limits_{\underset{i \neq j}{i,{j = 1}}}^{L/2}{\left( {D_{i}^{H}D_{j}} \right) \otimes \left( {H_{i}^{H}H_{j}} \right)^{*}}} \\        {= {{\sum\limits_{i = 1}^{L/2}{I_{p} \otimes \left( {H_{i}^{H}H_{i}} \right)}} + {\sum\limits_{\underset{i \neq j}{i,{j = 1}}}^{L/2}{\left( {C_{i}^{H}C_{j}} \right) \otimes \left( {H_{i}^{H}H_{j}} \right)}} +}} \\        {\sum\limits_{\underset{i \neq j}{i,{j = 1}}}^{L/2}{\left( {D_{i}^{H}D_{j}} \right) \otimes \left( {H_{i}^{H}H_{j}} \right)^{*}}}        \end{matrix} & (113)        \end{matrix}$

Usually, H⁽¹⁾ has nonzero entries in all positions. Therefore amulti-user detector, which may include de-correlating anddecision-feedback, should be employed for data estimation. However, ifrelations (102) or (103) (above) could be exploited, the number ofnonzero entries in H⁽¹⁾ can be halved. This will significantly reducethe cross-interference terms, thus improving the detector performance.

In this case we assume D_(k)=0 for any k, or C_(k)=0 for any k. To halvethe nonzero entries in H⁽¹⁾, the following must hold for any (i,j) pair,wherein relation (108) is applicable:C _(i) ^(H) C _(j) =C _(j) ^(H) C _(i), (∀i≠j)  (114)D _(i) ^(H) D _(j) =D _(j) ^(H) D _(i), (∀i≠j).  (115)

In an alternative embodiment, we assume that both D_(k) and C_(k) arenonzero matrices. Relation (109) is applicable if:C _(i) ^(H) C _(j)=−(D _(j) ^(H) D _(i)), (∀i≠j).  (116)

The number of nonzero entries in H⁽¹⁾ can be halved if relation (116)holds for any (i,j) pair.

In addition to the examples for recovering the transmitted symbolsanother example may be found in found in co-pending U.S. patentapplication Ser. No. 09/539,819 filed on Mar. 31, 2000 entitledSPACE-TIME CODE FOR MULTIPLE ANTENNA TRANSMISSION, assigned to assigneeof the present invention and incorporated herein by reference. Theapplication provides a system to recover transmitted symbols using amaximum likelihood sequence estimator (MLSE) decoder implemented withthe Viterbi algorithm with a decoding trellis.

Let us consider the four-antenna case. It is easy to see that, forexample, the following pair of matrices verify relation (114):$\begin{matrix}{{C_{12} = {{\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}\quad{and}\quad C_{34}} = \begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}}},{or}} & (117) \\{C_{12} = {{\begin{bmatrix}1 & 0 \\0 & j\end{bmatrix}\quad{and}\quad C_{34}} = {\begin{bmatrix}0 & 1 \\j & 0\end{bmatrix}.}}} & (118)\end{matrix}$

Using relation (117) results in: $\begin{matrix}{{{{H = \begin{bmatrix}H_{12} & H_{34} \\H_{34} & H_{12}\end{bmatrix}};{H^{(1)} = \begin{bmatrix}a & 0 & b & 0 \\0 & a & 0 & b \\b & 0 & a & 0 \\0 & b & 0 & a\end{bmatrix}}},{a = {{h_{1}}^{2} + {h_{2}}^{2} + {h_{3}}^{2} + {h_{4}}^{2}}}}{{b = {x + x^{*}}},{x\quad{given}\quad{in}\quad(107)}}{y^{(1)} = {{H^{H}r} = {{H^{(1)}s} + {H^{H}n}}}}} & (119)\end{matrix}$

Note that H⁽¹⁾ is circular matrix and denoting: $\begin{matrix}{H^{(2)} = \begin{bmatrix}a & 0 & {- b} & 0 \\0 & a & 0 & {- b} \\{- b} & 0 & a & 0 \\0 & {- b} & 0 & a\end{bmatrix}} & (120)\end{matrix}$

-   -   results in:        H ⁽²⁾ H ^(H) r=H ⁽²⁾ H ^(H)(Hs+n)=H ⁽²⁾ H ⁽¹⁾ s+H ⁽²⁾ H ^(H)        n=(a ² −b ²)I ₄ s+H ⁽²⁾ H ^(H) n  (121)

Applying sign(.) function in relation (121) gives the data estimates ofs. Thus, the detection is very simple but the noise is enhanced [see(121)]. However, these data estimates can be further improved using amultistage interference cancellation (MIC) scheme. Relation (121)provides the first data estimates (first stage). The next stage(s) canuse y⁽¹⁾ from relation (119) and the previous stage data estimations tocancel the interference given by the b elements in H⁽¹⁾ the matrix. Thecomputation overhead introduced by MIC is insignificant because H⁽¹⁾ isa sparse matrix. A sparse matrix is a matrix where the number of nonzeroelements situated off the main diagonal is less than the number of zeroelements. Also, b is a real value thus, multiplication with ±1±j is as amatter of fact a change of sign. The estimation can be performed usinghard bit or soft bit decision. The algorithm can provide soft outputs,which is desirable for the next processing blocks, (e.g. convolutionaldecoder).

An example of a scheme that includes the D_(2k−1,2k) matrices [see(110)] is: $\begin{matrix}{H = \begin{bmatrix}H_{12} & H_{34} \\{- H_{34}^{*}} & H_{12}^{*}\end{bmatrix}} & (122)\end{matrix}$

For this scheme, (109) is applicable while the detection complexity iskept at the same level as in the previous case.

Some remarks regarding the proposed class of STC are underlinedsubsequently.

-   -   There is an intrinsic relation between the Z matrix that        provides the STC and its equivalent representation in the H        matrix. The design has been carried out using the H matrix        because the target was a simple receiver.    -   The STC design presented herein is quite general and        straightforward. Also, can provide a rate R=1 STC.    -   Different STC schemes of the proposed class can be concatenated,        i.e. a sequence generated by relations (119) and (122),        alternately.

Embodiments of the invention has been shown as exemplars. Alternativeand modifications by those skilled in the art are deemed to be in thespirit and scope of the invention. For example, consider the following.

It is well known that there exist several equivalent versions of theAlamouti code. For example, the code $\begin{matrix}{C_{A\quad L\quad{A2}} = \begin{bmatrix}s_{1} & s_{2} \\s_{2}^{*} & {- s_{1}^{*}}\end{bmatrix}} & (123)\end{matrix}$

-   -   has exactly the same decoding properties as (108) and (109).        Thus, other embodiments of code structures may use these        Alamouti versions, (e.g. (123), as sub-blocks to construct        structured matrix classes).

Another way to build embodiments of the invention is to operatepermutations, which may be column-wise or row-wise, on a given STCstructure, as in the following example, wherein the step of permuting isdenoted by the arrow $\begin{matrix}{S = {{\left\lbrack \quad\begin{matrix}s_{1} & s_{2} & s_{3} & s_{4} \\{- s_{2}^{*}} & s_{1}^{*} & {- s_{4}^{*}} & {- s_{3}^{*}} \\s_{3} & s_{4} & s_{1} & s_{2} \\{- s_{4}^{*}} & {- s_{3}^{*}} & {- s_{2}^{*}} & {- s_{1}^{*}}\end{matrix}\quad \right\rbrack->S} = \left\lbrack \quad\begin{matrix}s_{1} & s_{2} & s_{3} & s_{4} \\s_{3} & s_{4} & s_{1} & s_{2} \\{- s_{2}^{*}} & s_{1}^{*} & {- s_{4}^{*}} & {- s_{3}^{*}} \\{- s_{4}^{*}} & {- s_{3}^{*}} & {- s_{2}^{*}} & {- s_{1}^{*}}\end{matrix}\quad \right\rbrack}} & (124)\end{matrix}$

-   -   where the second and the third row have been permuted. Note that        the first S matrix in (124) is the counterpart of the H matrix        in (119).

Also, deterministic phase rotations can be applied to antennas. Recallthat the columns are associated with the antennas. Therefore if a phaserotation is applied to all the elements of a given column, this willtranslate as a phase rotation to the corresponding channel, andimplicitly to the corresponding antenna. This can be useful to combatthe degradation due to the fading.

As will be recognized by those skilled in the art, the innovativeconcepts described in the present application can be modified and variedover a tremendous range of applications, and accordingly the scope ofpatented subject matter is not limited by any of the specific exemplaryteachings given.

1. A method of transmitting a signal over a plurality of antennas comprising the steps of, creating a family of structured matrix classes comprising complex elements; selecting a structured matrix class of said family of structured matrix classes; selecting a block of channel symbols; selecting a code matrix comprising a plurality of rows and a plurality of columns from said structured matrix class to be applied to said plurality of antennas; associating each of said columns of said plurality of columns of said code matrix to an antenna of said plurality of antennas; associating a row of said plurality of rows of said code matrix to cover a time duration of said selected block of data symbols; and transmitting using said associated antenna each said column of said code matrix over said corresponding time duration of said selected block of channel symbols.
 2. A method of transmitting a signal consisting of K channel symbols, complex conjugates of the symbols, and negative complex conjugates of the symbols over a plurality of antennas comprising the steps of: selecting a first portion of K/2 channel symbols; creating a plurality of first matrices comprising the selected first portion of K/2 channel symbols, their complex conjugates, and their negative complex conjugates; selecting a second portion of K/2 channel symbols; creating a plurality of second matrices comprising the selected second portion of K/2 channel symbols, their complex conjugates, and their negative complex conjugates; creating a code matrix having a diagonal and an anti-diagonal said code matrix comprising: the plurality of said first matrices on the diagonal of said code matrix; and the plurality of said second matrices on the anti-diagonal of said code matrix; transmitting the channel symbols, their complex conjugates, and their negative complex conjugates via a plurality of antennas.
 3. An arrangement for transmitting a signal consisting of symbols comprising: a coder for coding complex symbols to channel symbols in blocks having the length of a given K; and means for transmitting the channel symbols via several different channels and plurality of antennas; wherein the coder is arranged to code the symbols by: creating a family of structured matrix classes comprising complex elements; selecting a structured matrix class of said family of structured matrix classes; selecting a block of channel symbols; selecting a code matrix comprising a plurality of rows and a plurality of columns from said structured matrix class to be applied to said plurality of antennas; associating each of said columns of said plurality of columns of said code matrix to an antenna of said plurality of antennas; associating a row of said plurality of rows of said code matrix to cover a time duration of said selected block of data symbols; and selecting the channel symbols from the code matrix.
 4. An arrangement for transmitting a signal consisting of symbols, complex conjugates of the symbols, and negative complex conjugates of the symbols over a plurality of antennas comprising: a coder for coding complex symbols to channel symbols in blocks having the length of a given K; and means for transmitting the channel symbols via several different channels and a plurality of antennas; wherein the coder is arranged to code the symbols by: selecting a first portion of K/2 channel symbols; creating a plurality of first matrices comprising the selected first portion of K/2 channel symbols, their complex conjugates, and their negative complex conjugates; selecting a second portion of K/2 channel symbols; creating a plurality of second matrices comprising the selected second portion of K/2 channel symbols, their complex conjugates, and their negative complex conjugates; creating a code matrix having a diagonal and an anti-diagonal, said code matrix comprising: a plurality of said first matrices on the diagonal of said code matrix; a plurality of said second matrices on the anil-diagonal of said code matrix; and selecting the channel symbols from the code matrix.
 5. In a communication system wherein a digital signal consisting of symbols coded to channel symbols in blocks of given length and transmitted via several different channels and two or more antennas, and wherein the coding is defined by a code matrix comprising an orthogonal block code, in that information is transmitted in redundant directions of the orthogonal block code such that a code rate is higher than what is allowed by orthogonality, the method comprising the steps of: decoding the information transmitted in the redundant directions; and decoding the information transmitted in other directions after the information transmitted in the redundant directions has been decoded.
 6. The method of claim 5, wherein before the decoding, interference caused by the information transmitted in the redundant directions is suppressed from the information transmitted in other directions.
 7. A receiver in a communication system comprising: means for receiving channel symbols coded blocks of given length, wherein the coding is defined by a code matrix comprising an orthogonal block code, and wherein the channel symbols were transmitted via several different channels and two or more antennas including information in redundant directions of the orthogonal block code; and means decoding the information transmitted in the redundant directions first and the information transmitted in other directions next.
 8. The receiver of claim 7, further comprising means for suppressing the interference caused by the information transmitted in the redundant directions from the information transmitted in other directions, before the decoding is performed. 